Algebra Examples

$1 + \cos (18 x)$

Step 1

Simplify each term.

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Step 1.1

Factor $2$ out of $18 x$ .

$1 + \cos (2 (9 x))$

Step 1.2

Simplify each term.

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Step 1.2.1

Factor $3$ out of $9 x$ .

$1 + 2 \cos^{2} (3 (3 x)) - 1$

Step 1.2.2

Simplify each term.

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Step 1.2.2.1

Use the triple-angle identity to transform $\cos (3 x)$ to $4 \cos^{3} (x) - 3 \cos (x)$ .

$1 + 2 {(4 {(4 \cos^{3} (x) - 3 \cos (x))}^{3} - 3 \cos (3 x))}^{2} - 1$

Step 1.2.2.2

Use the Binomial Theorem.

$1 + 2 {(4 ({(4 \cos^{3} (x))}^{3} + 3 {(4 \cos^{3} (x))}^{2} (- 3 \cos (x)) + 3 (4 \cos^{3} (x)) {(- 3 \cos (x))}^{2} + {(- 3 \cos (x))}^{3}) - 3 \cos (3 x))}^{2} - 1$

Step 1.2.2.3

Simplify each term.

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Step 1.2.2.3.1

Apply the product rule to $4 \cos^{3} (x)$ .

$1 + 2 {(4 (4^{3} {(\cos^{3} (x))}^{3} + 3 {(4 \cos^{3} (x))}^{2} (- 3 \cos (x)) + 3 (4 \cos^{3} (x)) {(- 3 \cos (x))}^{2} + {(- 3 \cos (x))}^{3}) - 3 \cos (3 x))}^{2} - 1$

Step 1.2.2.3.2

Raise $4$ to the power of $3$ .

$1 + 2 {(4 (64 {(\cos^{3} (x))}^{3} + 3 {(4 \cos^{3} (x))}^{2} (- 3 \cos (x)) + 3 (4 \cos^{3} (x)) {(- 3 \cos (x))}^{2} + {(- 3 \cos (x))}^{3}) - 3 \cos (3 x))}^{2} - 1$

Step 1.2.2.3.3

Multiply the exponents in ${(\cos^{3} (x))}^{3}$ .

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Step 1.2.2.3.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$1 + 2 {(4 (64 {\cos (x)}^{3 \cdot 3} + 3 {(4 \cos^{3} (x))}^{2} (- 3 \cos (x)) + 3 (4 \cos^{3} (x)) {(- 3 \cos (x))}^{2} + {(- 3 \cos (x))}^{3}) - 3 \cos (3 x))}^{2} - 1$

Step 1.2.2.3.3.2

Multiply $3$ by $3$ .

$1 + 2 {(4 (64 \cos^{9} (x) + 3 {(4 \cos^{3} (x))}^{2} (- 3 \cos (x)) + 3 (4 \cos^{3} (x)) {(- 3 \cos (x))}^{2} + {(- 3 \cos (x))}^{3}) - 3 \cos (3 x))}^{2} - 1$

Step 1.2.2.3.4

Apply the product rule to $4 \cos^{3} (x)$ .

$1 + 2 {(4 (64 \cos^{9} (x) + 3 (4^{2} {(\cos^{3} (x))}^{2}) (- 3 \cos (x)) + 3 (4 \cos^{3} (x)) {(- 3 \cos (x))}^{2} + {(- 3 \cos (x))}^{3}) - 3 \cos (3 x))}^{2} - 1$

Step 1.2.2.3.5

Raise $4$ to the power of $2$ .

$1 + 2 {(4 (64 \cos^{9} (x) + 3 (16 {(\cos^{3} (x))}^{2}) (- 3 \cos (x)) + 3 (4 \cos^{3} (x)) {(- 3 \cos (x))}^{2} + {(- 3 \cos (x))}^{3}) - 3 \cos (3 x))}^{2} - 1$

Step 1.2.2.3.6

Multiply the exponents in ${(\cos^{3} (x))}^{2}$ .

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Step 1.2.2.3.6.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$1 + 2 {(4 (64 \cos^{9} (x) + 3 (16 {\cos (x)}^{3 \cdot 2}) (- 3 \cos (x)) + 3 (4 \cos^{3} (x)) {(- 3 \cos (x))}^{2} + {(- 3 \cos (x))}^{3}) - 3 \cos (3 x))}^{2} - 1$

Step 1.2.2.3.6.2

Multiply $3$ by $2$ .

$1 + 2 {(4 (64 \cos^{9} (x) + 3 (16 \cos^{6} (x)) (- 3 \cos (x)) + 3 (4 \cos^{3} (x)) {(- 3 \cos (x))}^{2} + {(- 3 \cos (x))}^{3}) - 3 \cos (3 x))}^{2} - 1$

Step 1.2.2.3.7

Multiply $\cos^{6} (x)$ by $\cos (x)$ by adding the exponents.

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Step 1.2.2.3.7.1

Move $\cos (x)$ .

$1 + 2 {(4 (64 \cos^{9} (x) + 3 (16 (\cos (x) \cos^{6} (x))) \cdot - 3 + 3 (4 \cos^{3} (x)) {(- 3 \cos (x))}^{2} + {(- 3 \cos (x))}^{3}) - 3 \cos (3 x))}^{2} - 1$

Step 1.2.2.3.7.2

Multiply $\cos (x)$ by $\cos^{6} (x)$ .

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Step 1.2.2.3.7.2.1

Raise $\cos (x)$ to the power of $1$ .

$1 + 2 {(4 (64 \cos^{9} (x) + 3 (16 (\cos^{1} (x) \cos^{6} (x))) \cdot - 3 + 3 (4 \cos^{3} (x)) {(- 3 \cos (x))}^{2} + {(- 3 \cos (x))}^{3}) - 3 \cos (3 x))}^{2} - 1$

Step 1.2.2.3.7.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$1 + 2 {(4 (64 \cos^{9} (x) + 3 (16 {\cos (x)}^{1 + 6}) \cdot - 3 + 3 (4 \cos^{3} (x)) {(- 3 \cos (x))}^{2} + {(- 3 \cos (x))}^{3}) - 3 \cos (3 x))}^{2} - 1$

Step 1.2.2.3.7.3

Add $1$ and $6$ .

$1 + 2 {(4 (64 \cos^{9} (x) + 3 (16 \cos^{7} (x)) \cdot - 3 + 3 (4 \cos^{3} (x)) {(- 3 \cos (x))}^{2} + {(- 3 \cos (x))}^{3}) - 3 \cos (3 x))}^{2} - 1$

Step 1.2.2.3.8

Multiply $16$ by $3$ .

$1 + 2 {(4 (64 \cos^{9} (x) + 48 \cos^{7} (x) \cdot - 3 + 3 (4 \cos^{3} (x)) {(- 3 \cos (x))}^{2} + {(- 3 \cos (x))}^{3}) - 3 \cos (3 x))}^{2} - 1$

Step 1.2.2.3.9

Multiply $- 3$ by $48$ .

$1 + 2 {(4 (64 \cos^{9} (x) - 144 \cos^{7} (x) + 3 (4 \cos^{3} (x)) {(- 3 \cos (x))}^{2} + {(- 3 \cos (x))}^{3}) - 3 \cos (3 x))}^{2} - 1$

Step 1.2.2.3.10

Multiply $4$ by $3$ .

$1 + 2 {(4 (64 \cos^{9} (x) - 144 \cos^{7} (x) + 12 \cos^{3} (x) {(- 3 \cos (x))}^{2} + {(- 3 \cos (x))}^{3}) - 3 \cos (3 x))}^{2} - 1$

Step 1.2.2.3.11

Apply the product rule to $- 3 \cos (x)$ .

$1 + 2 {(4 (64 \cos^{9} (x) - 144 \cos^{7} (x) + 12 \cos^{3} (x) ({(- 3)}^{2} \cos^{2} (x)) + {(- 3 \cos (x))}^{3}) - 3 \cos (3 x))}^{2} - 1$

Step 1.2.2.3.12

Rewrite using the commutative property of multiplication.

$1 + 2 {(4 (64 \cos^{9} (x) - 144 \cos^{7} (x) + 12 \cdot {(- 3)}^{2} \cos^{3} (x) \cos^{2} (x) + {(- 3 \cos (x))}^{3}) - 3 \cos (3 x))}^{2} - 1$

Step 1.2.2.3.13

Multiply $\cos^{3} (x)$ by $\cos^{2} (x)$ by adding the exponents.

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Step 1.2.2.3.13.1

Move $\cos^{2} (x)$ .

$1 + 2 {(4 (64 \cos^{9} (x) - 144 \cos^{7} (x) + 12 \cdot {(- 3)}^{2} (\cos^{2} (x) \cos^{3} (x)) + {(- 3 \cos (x))}^{3}) - 3 \cos (3 x))}^{2} - 1$

Step 1.2.2.3.13.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$1 + 2 {(4 (64 \cos^{9} (x) - 144 \cos^{7} (x) + 12 \cdot {(- 3)}^{2} {\cos (x)}^{2 + 3} + {(- 3 \cos (x))}^{3}) - 3 \cos (3 x))}^{2} - 1$

Step 1.2.2.3.13.3

Add $2$ and $3$ .

$1 + 2 {(4 (64 \cos^{9} (x) - 144 \cos^{7} (x) + 12 \cdot {(- 3)}^{2} \cos^{5} (x) + {(- 3 \cos (x))}^{3}) - 3 \cos (3 x))}^{2} - 1$

Step 1.2.2.3.14

Raise $- 3$ to the power of $2$ .

$1 + 2 {(4 (64 \cos^{9} (x) - 144 \cos^{7} (x) + 12 \cdot 9 \cos^{5} (x) + {(- 3 \cos (x))}^{3}) - 3 \cos (3 x))}^{2} - 1$

Step 1.2.2.3.15

Multiply $12$ by $9$ .

$1 + 2 {(4 (64 \cos^{9} (x) - 144 \cos^{7} (x) + 108 \cos^{5} (x) + {(- 3 \cos (x))}^{3}) - 3 \cos (3 x))}^{2} - 1$

Step 1.2.2.3.16

Apply the product rule to $- 3 \cos (x)$ .

$1 + 2 {(4 (64 \cos^{9} (x) - 144 \cos^{7} (x) + 108 \cos^{5} (x) + {(- 3)}^{3} \cos^{3} (x)) - 3 \cos (3 x))}^{2} - 1$

Step 1.2.2.3.17

Raise $- 3$ to the power of $3$ .

$1 + 2 {(4 (64 \cos^{9} (x) - 144 \cos^{7} (x) + 108 \cos^{5} (x) - 27 \cos^{3} (x)) - 3 \cos (3 x))}^{2} - 1$

Step 1.2.2.4

Apply the distributive property.

$1 + 2 {(4 (64 \cos^{9} (x)) + 4 (- 144 \cos^{7} (x)) + 4 (108 \cos^{5} (x)) + 4 (- 27 \cos^{3} (x)) - 3 \cos (3 x))}^{2} - 1$

Step 1.2.2.5

Simplify.

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Step 1.2.2.5.1

Multiply $64$ by $4$ .

$1 + 2 {(256 \cos^{9} (x) + 4 (- 144 \cos^{7} (x)) + 4 (108 \cos^{5} (x)) + 4 (- 27 \cos^{3} (x)) - 3 \cos (3 x))}^{2} - 1$

Step 1.2.2.5.2

Multiply $- 144$ by $4$ .

$1 + 2 {(256 \cos^{9} (x) - 576 \cos^{7} (x) + 4 (108 \cos^{5} (x)) + 4 (- 27 \cos^{3} (x)) - 3 \cos (3 x))}^{2} - 1$

Step 1.2.2.5.3

Multiply $108$ by $4$ .

$1 + 2 {(256 \cos^{9} (x) - 576 \cos^{7} (x) + 432 \cos^{5} (x) + 4 (- 27 \cos^{3} (x)) - 3 \cos (3 x))}^{2} - 1$

Step 1.2.2.5.4

Multiply $- 27$ by $4$ .

$1 + 2 {(256 \cos^{9} (x) - 576 \cos^{7} (x) + 432 \cos^{5} (x) - 108 \cos^{3} (x) - 3 \cos (3 x))}^{2} - 1$

Step 1.2.2.6

Use the triple-angle identity to transform $\cos (3 x)$ to $4 \cos^{3} (x) - 3 \cos (x)$ .

$1 + 2 {(256 \cos^{9} (x) - 576 \cos^{7} (x) + 432 \cos^{5} (x) - 108 \cos^{3} (x) - 3 (4 \cos^{3} (x) - 3 \cos (x)))}^{2} - 1$

Step 1.2.2.7

Apply the distributive property.

$1 + 2 {(256 \cos^{9} (x) - 576 \cos^{7} (x) + 432 \cos^{5} (x) - 108 \cos^{3} (x) - 3 (4 \cos^{3} (x)) - 3 (- 3 \cos (x)))}^{2} - 1$

Step 1.2.2.8

Multiply $4$ by $- 3$ .

$1 + 2 {(256 \cos^{9} (x) - 576 \cos^{7} (x) + 432 \cos^{5} (x) - 108 \cos^{3} (x) - 12 \cos^{3} (x) - 3 (- 3 \cos (x)))}^{2} - 1$

Step 1.2.2.9

Multiply $- 3$ by $- 3$ .

$1 + 2 {(256 \cos^{9} (x) - 576 \cos^{7} (x) + 432 \cos^{5} (x) - 108 \cos^{3} (x) - 12 \cos^{3} (x) + 9 \cos (x))}^{2} - 1$

Step 1.2.3

Subtract $12 \cos^{3} (x)$ from $- 108 \cos^{3} (x)$ .

$1 + 2 {(256 \cos^{9} (x) - 576 \cos^{7} (x) + 432 \cos^{5} (x) - 120 \cos^{3} (x) + 9 \cos (x))}^{2} - 1$

Step 1.2.4

Rewrite ${(256 \cos^{9} (x) - 576 \cos^{7} (x) + 432 \cos^{5} (x) - 120 \cos^{3} (x) + 9 \cos (x))}^{2}$ as $(256 \cos^{9} (x) - 576 \cos^{7} (x) + 432 \cos^{5} (x) - 120 \cos^{3} (x) + 9 \cos (x)) (256 \cos^{9} (x) - 576 \cos^{7} (x) + 432 \cos^{5} (x) - 120 \cos^{3} (x) + 9 \cos (x))$ .

$1 + 2 ((256 \cos^{9} (x) - 576 \cos^{7} (x) + 432 \cos^{5} (x) - 120 \cos^{3} (x) + 9 \cos (x)) (256 \cos^{9} (x) - 576 \cos^{7} (x) + 432 \cos^{5} (x) - 120 \cos^{3} (x) + 9 \cos (x))) - 1$

Step 1.2.5

Expand $(256 \cos^{9} (x) - 576 \cos^{7} (x) + 432 \cos^{5} (x) - 120 \cos^{3} (x) + 9 \cos (x)) (256 \cos^{9} (x) - 576 \cos^{7} (x) + 432 \cos^{5} (x) - 120 \cos^{3} (x) + 9 \cos (x))$ by multiplying each term in the first expression by each term in the second expression.

$1 + 2 (256 \cos^{9} (x) (256 \cos^{9} (x)) + 256 \cos^{9} (x) (- 576 \cos^{7} (x)) + 256 \cos^{9} (x) (432 \cos^{5} (x)) + 256 \cos^{9} (x) (- 120 \cos^{3} (x)) + 256 \cos^{9} (x) (9 \cos (x)) - 576 \cos^{7} (x) (256 \cos^{9} (x)) - 576 \cos^{7} (x) (- 576 \cos^{7} (x)) - 576 \cos^{7} (x) (432 \cos^{5} (x)) - 576 \cos^{7} (x) (- 120 \cos^{3} (x)) - 576 \cos^{7} (x) (9 \cos (x)) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6

Simplify each term.

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Step 1.2.6.1

Rewrite using the commutative property of multiplication.

$1 + 2 (256 \cdot 256 \cos^{9} (x) \cos^{9} (x) + 256 \cos^{9} (x) (- 576 \cos^{7} (x)) + 256 \cos^{9} (x) (432 \cos^{5} (x)) + 256 \cos^{9} (x) (- 120 \cos^{3} (x)) + 256 \cos^{9} (x) (9 \cos (x)) - 576 \cos^{7} (x) (256 \cos^{9} (x)) - 576 \cos^{7} (x) (- 576 \cos^{7} (x)) - 576 \cos^{7} (x) (432 \cos^{5} (x)) - 576 \cos^{7} (x) (- 120 \cos^{3} (x)) - 576 \cos^{7} (x) (9 \cos (x)) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.2

Multiply $\cos^{9} (x)$ by $\cos^{9} (x)$ by adding the exponents.

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Step 1.2.6.2.1

Move $\cos^{9} (x)$ .

$1 + 2 (256 \cdot 256 (\cos^{9} (x) \cos^{9} (x)) + 256 \cos^{9} (x) (- 576 \cos^{7} (x)) + 256 \cos^{9} (x) (432 \cos^{5} (x)) + 256 \cos^{9} (x) (- 120 \cos^{3} (x)) + 256 \cos^{9} (x) (9 \cos (x)) - 576 \cos^{7} (x) (256 \cos^{9} (x)) - 576 \cos^{7} (x) (- 576 \cos^{7} (x)) - 576 \cos^{7} (x) (432 \cos^{5} (x)) - 576 \cos^{7} (x) (- 120 \cos^{3} (x)) - 576 \cos^{7} (x) (9 \cos (x)) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$1 + 2 (256 \cdot 256 {\cos (x)}^{9 + 9} + 256 \cos^{9} (x) (- 576 \cos^{7} (x)) + 256 \cos^{9} (x) (432 \cos^{5} (x)) + 256 \cos^{9} (x) (- 120 \cos^{3} (x)) + 256 \cos^{9} (x) (9 \cos (x)) - 576 \cos^{7} (x) (256 \cos^{9} (x)) - 576 \cos^{7} (x) (- 576 \cos^{7} (x)) - 576 \cos^{7} (x) (432 \cos^{5} (x)) - 576 \cos^{7} (x) (- 120 \cos^{3} (x)) - 576 \cos^{7} (x) (9 \cos (x)) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.2.3

Add $9$ and $9$ .

$1 + 2 (256 \cdot 256 \cos^{18} (x) + 256 \cos^{9} (x) (- 576 \cos^{7} (x)) + 256 \cos^{9} (x) (432 \cos^{5} (x)) + 256 \cos^{9} (x) (- 120 \cos^{3} (x)) + 256 \cos^{9} (x) (9 \cos (x)) - 576 \cos^{7} (x) (256 \cos^{9} (x)) - 576 \cos^{7} (x) (- 576 \cos^{7} (x)) - 576 \cos^{7} (x) (432 \cos^{5} (x)) - 576 \cos^{7} (x) (- 120 \cos^{3} (x)) - 576 \cos^{7} (x) (9 \cos (x)) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.3

Multiply $256$ by $256$ .

$1 + 2 (65536 \cos^{18} (x) + 256 \cos^{9} (x) (- 576 \cos^{7} (x)) + 256 \cos^{9} (x) (432 \cos^{5} (x)) + 256 \cos^{9} (x) (- 120 \cos^{3} (x)) + 256 \cos^{9} (x) (9 \cos (x)) - 576 \cos^{7} (x) (256 \cos^{9} (x)) - 576 \cos^{7} (x) (- 576 \cos^{7} (x)) - 576 \cos^{7} (x) (432 \cos^{5} (x)) - 576 \cos^{7} (x) (- 120 \cos^{3} (x)) - 576 \cos^{7} (x) (9 \cos (x)) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.4

Rewrite using the commutative property of multiplication.

$1 + 2 (65536 \cos^{18} (x) + 256 \cdot - 576 \cos^{9} (x) \cos^{7} (x) + 256 \cos^{9} (x) (432 \cos^{5} (x)) + 256 \cos^{9} (x) (- 120 \cos^{3} (x)) + 256 \cos^{9} (x) (9 \cos (x)) - 576 \cos^{7} (x) (256 \cos^{9} (x)) - 576 \cos^{7} (x) (- 576 \cos^{7} (x)) - 576 \cos^{7} (x) (432 \cos^{5} (x)) - 576 \cos^{7} (x) (- 120 \cos^{3} (x)) - 576 \cos^{7} (x) (9 \cos (x)) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.5

Multiply $\cos^{9} (x)$ by $\cos^{7} (x)$ by adding the exponents.

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Step 1.2.6.5.1

Move $\cos^{7} (x)$ .

$1 + 2 (65536 \cos^{18} (x) + 256 \cdot - 576 (\cos^{7} (x) \cos^{9} (x)) + 256 \cos^{9} (x) (432 \cos^{5} (x)) + 256 \cos^{9} (x) (- 120 \cos^{3} (x)) + 256 \cos^{9} (x) (9 \cos (x)) - 576 \cos^{7} (x) (256 \cos^{9} (x)) - 576 \cos^{7} (x) (- 576 \cos^{7} (x)) - 576 \cos^{7} (x) (432 \cos^{5} (x)) - 576 \cos^{7} (x) (- 120 \cos^{3} (x)) - 576 \cos^{7} (x) (9 \cos (x)) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.5.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$1 + 2 (65536 \cos^{18} (x) + 256 \cdot - 576 {\cos (x)}^{7 + 9} + 256 \cos^{9} (x) (432 \cos^{5} (x)) + 256 \cos^{9} (x) (- 120 \cos^{3} (x)) + 256 \cos^{9} (x) (9 \cos (x)) - 576 \cos^{7} (x) (256 \cos^{9} (x)) - 576 \cos^{7} (x) (- 576 \cos^{7} (x)) - 576 \cos^{7} (x) (432 \cos^{5} (x)) - 576 \cos^{7} (x) (- 120 \cos^{3} (x)) - 576 \cos^{7} (x) (9 \cos (x)) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.5.3

Add $7$ and $9$ .

$1 + 2 (65536 \cos^{18} (x) + 256 \cdot - 576 \cos^{16} (x) + 256 \cos^{9} (x) (432 \cos^{5} (x)) + 256 \cos^{9} (x) (- 120 \cos^{3} (x)) + 256 \cos^{9} (x) (9 \cos (x)) - 576 \cos^{7} (x) (256 \cos^{9} (x)) - 576 \cos^{7} (x) (- 576 \cos^{7} (x)) - 576 \cos^{7} (x) (432 \cos^{5} (x)) - 576 \cos^{7} (x) (- 120 \cos^{3} (x)) - 576 \cos^{7} (x) (9 \cos (x)) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.6

Multiply $256$ by $- 576$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 256 \cos^{9} (x) (432 \cos^{5} (x)) + 256 \cos^{9} (x) (- 120 \cos^{3} (x)) + 256 \cos^{9} (x) (9 \cos (x)) - 576 \cos^{7} (x) (256 \cos^{9} (x)) - 576 \cos^{7} (x) (- 576 \cos^{7} (x)) - 576 \cos^{7} (x) (432 \cos^{5} (x)) - 576 \cos^{7} (x) (- 120 \cos^{3} (x)) - 576 \cos^{7} (x) (9 \cos (x)) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.7

Rewrite using the commutative property of multiplication.

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 256 \cdot 432 \cos^{9} (x) \cos^{5} (x) + 256 \cos^{9} (x) (- 120 \cos^{3} (x)) + 256 \cos^{9} (x) (9 \cos (x)) - 576 \cos^{7} (x) (256 \cos^{9} (x)) - 576 \cos^{7} (x) (- 576 \cos^{7} (x)) - 576 \cos^{7} (x) (432 \cos^{5} (x)) - 576 \cos^{7} (x) (- 120 \cos^{3} (x)) - 576 \cos^{7} (x) (9 \cos (x)) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.8

Multiply $\cos^{9} (x)$ by $\cos^{5} (x)$ by adding the exponents.

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Step 1.2.6.8.1

Move $\cos^{5} (x)$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 256 \cdot 432 (\cos^{5} (x) \cos^{9} (x)) + 256 \cos^{9} (x) (- 120 \cos^{3} (x)) + 256 \cos^{9} (x) (9 \cos (x)) - 576 \cos^{7} (x) (256 \cos^{9} (x)) - 576 \cos^{7} (x) (- 576 \cos^{7} (x)) - 576 \cos^{7} (x) (432 \cos^{5} (x)) - 576 \cos^{7} (x) (- 120 \cos^{3} (x)) - 576 \cos^{7} (x) (9 \cos (x)) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.8.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 256 \cdot 432 {\cos (x)}^{5 + 9} + 256 \cos^{9} (x) (- 120 \cos^{3} (x)) + 256 \cos^{9} (x) (9 \cos (x)) - 576 \cos^{7} (x) (256 \cos^{9} (x)) - 576 \cos^{7} (x) (- 576 \cos^{7} (x)) - 576 \cos^{7} (x) (432 \cos^{5} (x)) - 576 \cos^{7} (x) (- 120 \cos^{3} (x)) - 576 \cos^{7} (x) (9 \cos (x)) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.8.3

Add $5$ and $9$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 256 \cdot 432 \cos^{14} (x) + 256 \cos^{9} (x) (- 120 \cos^{3} (x)) + 256 \cos^{9} (x) (9 \cos (x)) - 576 \cos^{7} (x) (256 \cos^{9} (x)) - 576 \cos^{7} (x) (- 576 \cos^{7} (x)) - 576 \cos^{7} (x) (432 \cos^{5} (x)) - 576 \cos^{7} (x) (- 120 \cos^{3} (x)) - 576 \cos^{7} (x) (9 \cos (x)) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.9

Multiply $256$ by $432$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) + 256 \cos^{9} (x) (- 120 \cos^{3} (x)) + 256 \cos^{9} (x) (9 \cos (x)) - 576 \cos^{7} (x) (256 \cos^{9} (x)) - 576 \cos^{7} (x) (- 576 \cos^{7} (x)) - 576 \cos^{7} (x) (432 \cos^{5} (x)) - 576 \cos^{7} (x) (- 120 \cos^{3} (x)) - 576 \cos^{7} (x) (9 \cos (x)) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.10

Rewrite using the commutative property of multiplication.

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) + 256 \cdot - 120 \cos^{9} (x) \cos^{3} (x) + 256 \cos^{9} (x) (9 \cos (x)) - 576 \cos^{7} (x) (256 \cos^{9} (x)) - 576 \cos^{7} (x) (- 576 \cos^{7} (x)) - 576 \cos^{7} (x) (432 \cos^{5} (x)) - 576 \cos^{7} (x) (- 120 \cos^{3} (x)) - 576 \cos^{7} (x) (9 \cos (x)) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.11

Multiply $\cos^{9} (x)$ by $\cos^{3} (x)$ by adding the exponents.

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Step 1.2.6.11.1

Move $\cos^{3} (x)$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) + 256 \cdot - 120 (\cos^{3} (x) \cos^{9} (x)) + 256 \cos^{9} (x) (9 \cos (x)) - 576 \cos^{7} (x) (256 \cos^{9} (x)) - 576 \cos^{7} (x) (- 576 \cos^{7} (x)) - 576 \cos^{7} (x) (432 \cos^{5} (x)) - 576 \cos^{7} (x) (- 120 \cos^{3} (x)) - 576 \cos^{7} (x) (9 \cos (x)) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.11.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) + 256 \cdot - 120 {\cos (x)}^{3 + 9} + 256 \cos^{9} (x) (9 \cos (x)) - 576 \cos^{7} (x) (256 \cos^{9} (x)) - 576 \cos^{7} (x) (- 576 \cos^{7} (x)) - 576 \cos^{7} (x) (432 \cos^{5} (x)) - 576 \cos^{7} (x) (- 120 \cos^{3} (x)) - 576 \cos^{7} (x) (9 \cos (x)) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.11.3

Add $3$ and $9$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) + 256 \cdot - 120 \cos^{12} (x) + 256 \cos^{9} (x) (9 \cos (x)) - 576 \cos^{7} (x) (256 \cos^{9} (x)) - 576 \cos^{7} (x) (- 576 \cos^{7} (x)) - 576 \cos^{7} (x) (432 \cos^{5} (x)) - 576 \cos^{7} (x) (- 120 \cos^{3} (x)) - 576 \cos^{7} (x) (9 \cos (x)) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.12

Multiply $256$ by $- 120$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 256 \cos^{9} (x) (9 \cos (x)) - 576 \cos^{7} (x) (256 \cos^{9} (x)) - 576 \cos^{7} (x) (- 576 \cos^{7} (x)) - 576 \cos^{7} (x) (432 \cos^{5} (x)) - 576 \cos^{7} (x) (- 120 \cos^{3} (x)) - 576 \cos^{7} (x) (9 \cos (x)) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.13

Multiply $\cos^{9} (x)$ by $\cos (x)$ by adding the exponents.

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Step 1.2.6.13.1

Move $\cos (x)$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 256 (\cos (x) \cos^{9} (x)) \cdot 9 - 576 \cos^{7} (x) (256 \cos^{9} (x)) - 576 \cos^{7} (x) (- 576 \cos^{7} (x)) - 576 \cos^{7} (x) (432 \cos^{5} (x)) - 576 \cos^{7} (x) (- 120 \cos^{3} (x)) - 576 \cos^{7} (x) (9 \cos (x)) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.13.2

Multiply $\cos (x)$ by $\cos^{9} (x)$ .

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Step 1.2.6.13.2.1

Raise $\cos (x)$ to the power of $1$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 256 (\cos^{1} (x) \cos^{9} (x)) \cdot 9 - 576 \cos^{7} (x) (256 \cos^{9} (x)) - 576 \cos^{7} (x) (- 576 \cos^{7} (x)) - 576 \cos^{7} (x) (432 \cos^{5} (x)) - 576 \cos^{7} (x) (- 120 \cos^{3} (x)) - 576 \cos^{7} (x) (9 \cos (x)) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.13.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 256 {\cos (x)}^{1 + 9} \cdot 9 - 576 \cos^{7} (x) (256 \cos^{9} (x)) - 576 \cos^{7} (x) (- 576 \cos^{7} (x)) - 576 \cos^{7} (x) (432 \cos^{5} (x)) - 576 \cos^{7} (x) (- 120 \cos^{3} (x)) - 576 \cos^{7} (x) (9 \cos (x)) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.13.3

Add $1$ and $9$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 256 \cos^{10} (x) \cdot 9 - 576 \cos^{7} (x) (256 \cos^{9} (x)) - 576 \cos^{7} (x) (- 576 \cos^{7} (x)) - 576 \cos^{7} (x) (432 \cos^{5} (x)) - 576 \cos^{7} (x) (- 120 \cos^{3} (x)) - 576 \cos^{7} (x) (9 \cos (x)) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.14

Multiply $9$ by $256$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 576 \cos^{7} (x) (256 \cos^{9} (x)) - 576 \cos^{7} (x) (- 576 \cos^{7} (x)) - 576 \cos^{7} (x) (432 \cos^{5} (x)) - 576 \cos^{7} (x) (- 120 \cos^{3} (x)) - 576 \cos^{7} (x) (9 \cos (x)) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.15

Rewrite using the commutative property of multiplication.

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 576 \cdot 256 \cos^{7} (x) \cos^{9} (x) - 576 \cos^{7} (x) (- 576 \cos^{7} (x)) - 576 \cos^{7} (x) (432 \cos^{5} (x)) - 576 \cos^{7} (x) (- 120 \cos^{3} (x)) - 576 \cos^{7} (x) (9 \cos (x)) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.16

Multiply $\cos^{7} (x)$ by $\cos^{9} (x)$ by adding the exponents.

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Step 1.2.6.16.1

Move $\cos^{9} (x)$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 576 \cdot 256 (\cos^{9} (x) \cos^{7} (x)) - 576 \cos^{7} (x) (- 576 \cos^{7} (x)) - 576 \cos^{7} (x) (432 \cos^{5} (x)) - 576 \cos^{7} (x) (- 120 \cos^{3} (x)) - 576 \cos^{7} (x) (9 \cos (x)) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.16.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 576 \cdot 256 {\cos (x)}^{9 + 7} - 576 \cos^{7} (x) (- 576 \cos^{7} (x)) - 576 \cos^{7} (x) (432 \cos^{5} (x)) - 576 \cos^{7} (x) (- 120 \cos^{3} (x)) - 576 \cos^{7} (x) (9 \cos (x)) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.16.3

Add $9$ and $7$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 576 \cdot 256 \cos^{16} (x) - 576 \cos^{7} (x) (- 576 \cos^{7} (x)) - 576 \cos^{7} (x) (432 \cos^{5} (x)) - 576 \cos^{7} (x) (- 120 \cos^{3} (x)) - 576 \cos^{7} (x) (9 \cos (x)) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.17

Multiply $- 576$ by $256$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) - 576 \cos^{7} (x) (- 576 \cos^{7} (x)) - 576 \cos^{7} (x) (432 \cos^{5} (x)) - 576 \cos^{7} (x) (- 120 \cos^{3} (x)) - 576 \cos^{7} (x) (9 \cos (x)) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.18

Rewrite using the commutative property of multiplication.

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) - 576 \cdot - 576 \cos^{7} (x) \cos^{7} (x) - 576 \cos^{7} (x) (432 \cos^{5} (x)) - 576 \cos^{7} (x) (- 120 \cos^{3} (x)) - 576 \cos^{7} (x) (9 \cos (x)) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.19

Multiply $\cos^{7} (x)$ by $\cos^{7} (x)$ by adding the exponents.

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Step 1.2.6.19.1

Move $\cos^{7} (x)$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) - 576 \cdot - 576 (\cos^{7} (x) \cos^{7} (x)) - 576 \cos^{7} (x) (432 \cos^{5} (x)) - 576 \cos^{7} (x) (- 120 \cos^{3} (x)) - 576 \cos^{7} (x) (9 \cos (x)) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.19.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) - 576 \cdot - 576 {\cos (x)}^{7 + 7} - 576 \cos^{7} (x) (432 \cos^{5} (x)) - 576 \cos^{7} (x) (- 120 \cos^{3} (x)) - 576 \cos^{7} (x) (9 \cos (x)) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.19.3

Add $7$ and $7$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) - 576 \cdot - 576 \cos^{14} (x) - 576 \cos^{7} (x) (432 \cos^{5} (x)) - 576 \cos^{7} (x) (- 120 \cos^{3} (x)) - 576 \cos^{7} (x) (9 \cos (x)) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.20

Multiply $- 576$ by $- 576$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 576 \cos^{7} (x) (432 \cos^{5} (x)) - 576 \cos^{7} (x) (- 120 \cos^{3} (x)) - 576 \cos^{7} (x) (9 \cos (x)) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.21

Rewrite using the commutative property of multiplication.

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 576 \cdot 432 \cos^{7} (x) \cos^{5} (x) - 576 \cos^{7} (x) (- 120 \cos^{3} (x)) - 576 \cos^{7} (x) (9 \cos (x)) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.22

Multiply $\cos^{7} (x)$ by $\cos^{5} (x)$ by adding the exponents.

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Step 1.2.6.22.1

Move $\cos^{5} (x)$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 576 \cdot 432 (\cos^{5} (x) \cos^{7} (x)) - 576 \cos^{7} (x) (- 120 \cos^{3} (x)) - 576 \cos^{7} (x) (9 \cos (x)) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.22.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 576 \cdot 432 {\cos (x)}^{5 + 7} - 576 \cos^{7} (x) (- 120 \cos^{3} (x)) - 576 \cos^{7} (x) (9 \cos (x)) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.22.3

Add $5$ and $7$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 576 \cdot 432 \cos^{12} (x) - 576 \cos^{7} (x) (- 120 \cos^{3} (x)) - 576 \cos^{7} (x) (9 \cos (x)) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.23

Multiply $- 576$ by $432$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) - 576 \cos^{7} (x) (- 120 \cos^{3} (x)) - 576 \cos^{7} (x) (9 \cos (x)) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.24

Rewrite using the commutative property of multiplication.

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) - 576 \cdot - 120 \cos^{7} (x) \cos^{3} (x) - 576 \cos^{7} (x) (9 \cos (x)) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.25

Multiply $\cos^{7} (x)$ by $\cos^{3} (x)$ by adding the exponents.

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Step 1.2.6.25.1

Move $\cos^{3} (x)$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) - 576 \cdot - 120 (\cos^{3} (x) \cos^{7} (x)) - 576 \cos^{7} (x) (9 \cos (x)) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.25.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) - 576 \cdot - 120 {\cos (x)}^{3 + 7} - 576 \cos^{7} (x) (9 \cos (x)) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.25.3

Add $3$ and $7$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) - 576 \cdot - 120 \cos^{10} (x) - 576 \cos^{7} (x) (9 \cos (x)) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.26

Multiply $- 576$ by $- 120$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 576 \cos^{7} (x) (9 \cos (x)) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.27

Multiply $\cos^{7} (x)$ by $\cos (x)$ by adding the exponents.

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Step 1.2.6.27.1

Move $\cos (x)$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 576 (\cos (x) \cos^{7} (x)) \cdot 9 + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.27.2

Multiply $\cos (x)$ by $\cos^{7} (x)$ .

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Step 1.2.6.27.2.1

Raise $\cos (x)$ to the power of $1$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 576 (\cos^{1} (x) \cos^{7} (x)) \cdot 9 + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.27.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 576 {\cos (x)}^{1 + 7} \cdot 9 + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.27.3

Add $1$ and $7$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 576 \cos^{8} (x) \cdot 9 + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.28

Multiply $9$ by $- 576$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 432 \cos^{5} (x) (256 \cos^{9} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.29

Rewrite using the commutative property of multiplication.

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 432 \cdot 256 \cos^{5} (x) \cos^{9} (x) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.30

Multiply $\cos^{5} (x)$ by $\cos^{9} (x)$ by adding the exponents.

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Step 1.2.6.30.1

Move $\cos^{9} (x)$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 432 \cdot 256 (\cos^{9} (x) \cos^{5} (x)) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.30.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 432 \cdot 256 {\cos (x)}^{9 + 5} + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.30.3

Add $9$ and $5$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 432 \cdot 256 \cos^{14} (x) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.31

Multiply $432$ by $256$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) + 432 \cos^{5} (x) (- 576 \cos^{7} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.32

Rewrite using the commutative property of multiplication.

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) + 432 \cdot - 576 \cos^{5} (x) \cos^{7} (x) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.33

Multiply $\cos^{5} (x)$ by $\cos^{7} (x)$ by adding the exponents.

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Step 1.2.6.33.1

Move $\cos^{7} (x)$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) + 432 \cdot - 576 (\cos^{7} (x) \cos^{5} (x)) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.33.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) + 432 \cdot - 576 {\cos (x)}^{7 + 5} + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.33.3

Add $7$ and $5$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) + 432 \cdot - 576 \cos^{12} (x) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.34

Multiply $432$ by $- 576$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 432 \cos^{5} (x) (432 \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.35

Rewrite using the commutative property of multiplication.

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 432 \cdot 432 \cos^{5} (x) \cos^{5} (x) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.36

Multiply $\cos^{5} (x)$ by $\cos^{5} (x)$ by adding the exponents.

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Step 1.2.6.36.1

Move $\cos^{5} (x)$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 432 \cdot 432 (\cos^{5} (x) \cos^{5} (x)) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.36.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 432 \cdot 432 {\cos (x)}^{5 + 5} + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.36.3

Add $5$ and $5$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 432 \cdot 432 \cos^{10} (x) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.37

Multiply $432$ by $432$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) + 432 \cos^{5} (x) (- 120 \cos^{3} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.38

Rewrite using the commutative property of multiplication.

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) + 432 \cdot - 120 \cos^{5} (x) \cos^{3} (x) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.39

Multiply $\cos^{5} (x)$ by $\cos^{3} (x)$ by adding the exponents.

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Step 1.2.6.39.1

Move $\cos^{3} (x)$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) + 432 \cdot - 120 (\cos^{3} (x) \cos^{5} (x)) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.39.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) + 432 \cdot - 120 {\cos (x)}^{3 + 5} + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.39.3

Add $3$ and $5$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) + 432 \cdot - 120 \cos^{8} (x) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.40

Multiply $432$ by $- 120$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 432 \cos^{5} (x) (9 \cos (x)) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.41

Multiply $\cos^{5} (x)$ by $\cos (x)$ by adding the exponents.

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Step 1.2.6.41.1

Move $\cos (x)$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 432 (\cos (x) \cos^{5} (x)) \cdot 9 - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.41.2

Multiply $\cos (x)$ by $\cos^{5} (x)$ .

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Step 1.2.6.41.2.1

Raise $\cos (x)$ to the power of $1$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 432 (\cos^{1} (x) \cos^{5} (x)) \cdot 9 - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.41.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 432 {\cos (x)}^{1 + 5} \cdot 9 - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.41.3

Add $1$ and $5$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 432 \cos^{6} (x) \cdot 9 - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.42

Multiply $9$ by $432$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 120 \cos^{3} (x) (256 \cos^{9} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.43

Rewrite using the commutative property of multiplication.

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 120 \cdot 256 \cos^{3} (x) \cos^{9} (x) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.44

Multiply $\cos^{3} (x)$ by $\cos^{9} (x)$ by adding the exponents.

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Step 1.2.6.44.1

Move $\cos^{9} (x)$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 120 \cdot 256 (\cos^{9} (x) \cos^{3} (x)) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.44.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 120 \cdot 256 {\cos (x)}^{9 + 3} - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.44.3

Add $9$ and $3$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 120 \cdot 256 \cos^{12} (x) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.45

Multiply $- 120$ by $256$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) - 120 \cos^{3} (x) (- 576 \cos^{7} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.46

Rewrite using the commutative property of multiplication.

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) - 120 \cdot - 576 \cos^{3} (x) \cos^{7} (x) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.47

Multiply $\cos^{3} (x)$ by $\cos^{7} (x)$ by adding the exponents.

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Step 1.2.6.47.1

Move $\cos^{7} (x)$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) - 120 \cdot - 576 (\cos^{7} (x) \cos^{3} (x)) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.47.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) - 120 \cdot - 576 {\cos (x)}^{7 + 3} - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.47.3

Add $7$ and $3$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) - 120 \cdot - 576 \cos^{10} (x) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.48

Multiply $- 120$ by $- 576$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) + 69120 \cos^{10} (x) - 120 \cos^{3} (x) (432 \cos^{5} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.49

Rewrite using the commutative property of multiplication.

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) + 69120 \cos^{10} (x) - 120 \cdot 432 \cos^{3} (x) \cos^{5} (x) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.50

Multiply $\cos^{3} (x)$ by $\cos^{5} (x)$ by adding the exponents.

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Step 1.2.6.50.1

Move $\cos^{5} (x)$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) + 69120 \cos^{10} (x) - 120 \cdot 432 (\cos^{5} (x) \cos^{3} (x)) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.50.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) + 69120 \cos^{10} (x) - 120 \cdot 432 {\cos (x)}^{5 + 3} - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.50.3

Add $5$ and $3$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) + 69120 \cos^{10} (x) - 120 \cdot 432 \cos^{8} (x) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.51

Multiply $- 120$ by $432$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) + 69120 \cos^{10} (x) - 51840 \cos^{8} (x) - 120 \cos^{3} (x) (- 120 \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.52

Rewrite using the commutative property of multiplication.

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) + 69120 \cos^{10} (x) - 51840 \cos^{8} (x) - 120 \cdot - 120 \cos^{3} (x) \cos^{3} (x) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.53

Multiply $\cos^{3} (x)$ by $\cos^{3} (x)$ by adding the exponents.

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Step 1.2.6.53.1

Move $\cos^{3} (x)$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) + 69120 \cos^{10} (x) - 51840 \cos^{8} (x) - 120 \cdot - 120 (\cos^{3} (x) \cos^{3} (x)) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.53.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) + 69120 \cos^{10} (x) - 51840 \cos^{8} (x) - 120 \cdot - 120 {\cos (x)}^{3 + 3} - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.53.3

Add $3$ and $3$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) + 69120 \cos^{10} (x) - 51840 \cos^{8} (x) - 120 \cdot - 120 \cos^{6} (x) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.54

Multiply $- 120$ by $- 120$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) + 69120 \cos^{10} (x) - 51840 \cos^{8} (x) + 14400 \cos^{6} (x) - 120 \cos^{3} (x) (9 \cos (x)) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.55

Multiply $\cos^{3} (x)$ by $\cos (x)$ by adding the exponents.

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Step 1.2.6.55.1

Move $\cos (x)$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) + 69120 \cos^{10} (x) - 51840 \cos^{8} (x) + 14400 \cos^{6} (x) - 120 (\cos (x) \cos^{3} (x)) \cdot 9 + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.55.2

Multiply $\cos (x)$ by $\cos^{3} (x)$ .

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Step 1.2.6.55.2.1

Raise $\cos (x)$ to the power of $1$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) + 69120 \cos^{10} (x) - 51840 \cos^{8} (x) + 14400 \cos^{6} (x) - 120 (\cos^{1} (x) \cos^{3} (x)) \cdot 9 + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.55.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) + 69120 \cos^{10} (x) - 51840 \cos^{8} (x) + 14400 \cos^{6} (x) - 120 {\cos (x)}^{1 + 3} \cdot 9 + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.55.3

Add $1$ and $3$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) + 69120 \cos^{10} (x) - 51840 \cos^{8} (x) + 14400 \cos^{6} (x) - 120 \cos^{4} (x) \cdot 9 + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.56

Multiply $9$ by $- 120$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) + 69120 \cos^{10} (x) - 51840 \cos^{8} (x) + 14400 \cos^{6} (x) - 1080 \cos^{4} (x) + 9 \cos (x) (256 \cos^{9} (x)) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.57

Multiply $\cos (x)$ by $\cos^{9} (x)$ by adding the exponents.

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Step 1.2.6.57.1

Move $\cos^{9} (x)$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) + 69120 \cos^{10} (x) - 51840 \cos^{8} (x) + 14400 \cos^{6} (x) - 1080 \cos^{4} (x) + 9 (\cos^{9} (x) \cos (x)) \cdot 256 + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.57.2

Multiply $\cos^{9} (x)$ by $\cos (x)$ .

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Step 1.2.6.57.2.1

Raise $\cos (x)$ to the power of $1$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) + 69120 \cos^{10} (x) - 51840 \cos^{8} (x) + 14400 \cos^{6} (x) - 1080 \cos^{4} (x) + 9 (\cos^{9} (x) \cos^{1} (x)) \cdot 256 + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.57.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) + 69120 \cos^{10} (x) - 51840 \cos^{8} (x) + 14400 \cos^{6} (x) - 1080 \cos^{4} (x) + 9 {\cos (x)}^{9 + 1} \cdot 256 + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.57.3

Add $9$ and $1$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) + 69120 \cos^{10} (x) - 51840 \cos^{8} (x) + 14400 \cos^{6} (x) - 1080 \cos^{4} (x) + 9 \cos^{10} (x) \cdot 256 + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.58

Multiply $256$ by $9$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) + 69120 \cos^{10} (x) - 51840 \cos^{8} (x) + 14400 \cos^{6} (x) - 1080 \cos^{4} (x) + 2304 \cos^{10} (x) + 9 \cos (x) (- 576 \cos^{7} (x)) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.59

Multiply $\cos (x)$ by $\cos^{7} (x)$ by adding the exponents.

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Step 1.2.6.59.1

Move $\cos^{7} (x)$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) + 69120 \cos^{10} (x) - 51840 \cos^{8} (x) + 14400 \cos^{6} (x) - 1080 \cos^{4} (x) + 2304 \cos^{10} (x) + 9 (\cos^{7} (x) \cos (x)) \cdot - 576 + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.59.2

Multiply $\cos^{7} (x)$ by $\cos (x)$ .

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Step 1.2.6.59.2.1

Raise $\cos (x)$ to the power of $1$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) + 69120 \cos^{10} (x) - 51840 \cos^{8} (x) + 14400 \cos^{6} (x) - 1080 \cos^{4} (x) + 2304 \cos^{10} (x) + 9 (\cos^{7} (x) \cos^{1} (x)) \cdot - 576 + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.59.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) + 69120 \cos^{10} (x) - 51840 \cos^{8} (x) + 14400 \cos^{6} (x) - 1080 \cos^{4} (x) + 2304 \cos^{10} (x) + 9 {\cos (x)}^{7 + 1} \cdot - 576 + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.59.3

Add $7$ and $1$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) + 69120 \cos^{10} (x) - 51840 \cos^{8} (x) + 14400 \cos^{6} (x) - 1080 \cos^{4} (x) + 2304 \cos^{10} (x) + 9 \cos^{8} (x) \cdot - 576 + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.60

Multiply $- 576$ by $9$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) + 69120 \cos^{10} (x) - 51840 \cos^{8} (x) + 14400 \cos^{6} (x) - 1080 \cos^{4} (x) + 2304 \cos^{10} (x) - 5184 \cos^{8} (x) + 9 \cos (x) (432 \cos^{5} (x)) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.61

Multiply $\cos (x)$ by $\cos^{5} (x)$ by adding the exponents.

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Step 1.2.6.61.1

Move $\cos^{5} (x)$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) + 69120 \cos^{10} (x) - 51840 \cos^{8} (x) + 14400 \cos^{6} (x) - 1080 \cos^{4} (x) + 2304 \cos^{10} (x) - 5184 \cos^{8} (x) + 9 (\cos^{5} (x) \cos (x)) \cdot 432 + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.61.2

Multiply $\cos^{5} (x)$ by $\cos (x)$ .

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Step 1.2.6.61.2.1

Raise $\cos (x)$ to the power of $1$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) + 69120 \cos^{10} (x) - 51840 \cos^{8} (x) + 14400 \cos^{6} (x) - 1080 \cos^{4} (x) + 2304 \cos^{10} (x) - 5184 \cos^{8} (x) + 9 (\cos^{5} (x) \cos^{1} (x)) \cdot 432 + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.61.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) + 69120 \cos^{10} (x) - 51840 \cos^{8} (x) + 14400 \cos^{6} (x) - 1080 \cos^{4} (x) + 2304 \cos^{10} (x) - 5184 \cos^{8} (x) + 9 {\cos (x)}^{5 + 1} \cdot 432 + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.61.3

Add $5$ and $1$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) + 69120 \cos^{10} (x) - 51840 \cos^{8} (x) + 14400 \cos^{6} (x) - 1080 \cos^{4} (x) + 2304 \cos^{10} (x) - 5184 \cos^{8} (x) + 9 \cos^{6} (x) \cdot 432 + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.62

Multiply $432$ by $9$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) + 69120 \cos^{10} (x) - 51840 \cos^{8} (x) + 14400 \cos^{6} (x) - 1080 \cos^{4} (x) + 2304 \cos^{10} (x) - 5184 \cos^{8} (x) + 3888 \cos^{6} (x) + 9 \cos (x) (- 120 \cos^{3} (x)) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.63

Multiply $\cos (x)$ by $\cos^{3} (x)$ by adding the exponents.

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Step 1.2.6.63.1

Move $\cos^{3} (x)$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) + 69120 \cos^{10} (x) - 51840 \cos^{8} (x) + 14400 \cos^{6} (x) - 1080 \cos^{4} (x) + 2304 \cos^{10} (x) - 5184 \cos^{8} (x) + 3888 \cos^{6} (x) + 9 (\cos^{3} (x) \cos (x)) \cdot - 120 + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.63.2

Multiply $\cos^{3} (x)$ by $\cos (x)$ .

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Step 1.2.6.63.2.1

Raise $\cos (x)$ to the power of $1$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) + 69120 \cos^{10} (x) - 51840 \cos^{8} (x) + 14400 \cos^{6} (x) - 1080 \cos^{4} (x) + 2304 \cos^{10} (x) - 5184 \cos^{8} (x) + 3888 \cos^{6} (x) + 9 (\cos^{3} (x) \cos^{1} (x)) \cdot - 120 + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.63.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) + 69120 \cos^{10} (x) - 51840 \cos^{8} (x) + 14400 \cos^{6} (x) - 1080 \cos^{4} (x) + 2304 \cos^{10} (x) - 5184 \cos^{8} (x) + 3888 \cos^{6} (x) + 9 {\cos (x)}^{3 + 1} \cdot - 120 + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.63.3

Add $3$ and $1$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) + 69120 \cos^{10} (x) - 51840 \cos^{8} (x) + 14400 \cos^{6} (x) - 1080 \cos^{4} (x) + 2304 \cos^{10} (x) - 5184 \cos^{8} (x) + 3888 \cos^{6} (x) + 9 \cos^{4} (x) \cdot - 120 + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.64

Multiply $- 120$ by $9$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) + 69120 \cos^{10} (x) - 51840 \cos^{8} (x) + 14400 \cos^{6} (x) - 1080 \cos^{4} (x) + 2304 \cos^{10} (x) - 5184 \cos^{8} (x) + 3888 \cos^{6} (x) - 1080 \cos^{4} (x) + 9 \cos (x) (9 \cos (x))) - 1$

Step 1.2.6.65

Multiply $9 \cos (x) (9 \cos (x))$ .

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Step 1.2.6.65.1

Multiply $9$ by $9$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) + 69120 \cos^{10} (x) - 51840 \cos^{8} (x) + 14400 \cos^{6} (x) - 1080 \cos^{4} (x) + 2304 \cos^{10} (x) - 5184 \cos^{8} (x) + 3888 \cos^{6} (x) - 1080 \cos^{4} (x) + 81 \cos (x) \cos (x)) - 1$

Step 1.2.6.65.2

Raise $\cos (x)$ to the power of $1$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) + 69120 \cos^{10} (x) - 51840 \cos^{8} (x) + 14400 \cos^{6} (x) - 1080 \cos^{4} (x) + 2304 \cos^{10} (x) - 5184 \cos^{8} (x) + 3888 \cos^{6} (x) - 1080 \cos^{4} (x) + 81 (\cos^{1} (x) \cos (x))) - 1$

Step 1.2.6.65.3

Raise $\cos (x)$ to the power of $1$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) + 69120 \cos^{10} (x) - 51840 \cos^{8} (x) + 14400 \cos^{6} (x) - 1080 \cos^{4} (x) + 2304 \cos^{10} (x) - 5184 \cos^{8} (x) + 3888 \cos^{6} (x) - 1080 \cos^{4} (x) + 81 (\cos^{1} (x) \cos^{1} (x))) - 1$

Step 1.2.6.65.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) + 69120 \cos^{10} (x) - 51840 \cos^{8} (x) + 14400 \cos^{6} (x) - 1080 \cos^{4} (x) + 2304 \cos^{10} (x) - 5184 \cos^{8} (x) + 3888 \cos^{6} (x) - 1080 \cos^{4} (x) + 81 {\cos (x)}^{1 + 1}) - 1$

Step 1.2.6.65.5

Add $1$ and $1$ .

$1 + 2 (65536 \cos^{18} (x) - 147456 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 147456 \cos^{16} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) + 69120 \cos^{10} (x) - 51840 \cos^{8} (x) + 14400 \cos^{6} (x) - 1080 \cos^{4} (x) + 2304 \cos^{10} (x) - 5184 \cos^{8} (x) + 3888 \cos^{6} (x) - 1080 \cos^{4} (x) + 81 \cos^{2} (x)) - 1$

Step 1.2.7

Subtract $147456 \cos^{16} (x)$ from $- 147456 \cos^{16} (x)$ .

$1 + 2 (65536 \cos^{18} (x) - 294912 \cos^{16} (x) + 110592 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) + 331776 \cos^{14} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) + 69120 \cos^{10} (x) - 51840 \cos^{8} (x) + 14400 \cos^{6} (x) - 1080 \cos^{4} (x) + 2304 \cos^{10} (x) - 5184 \cos^{8} (x) + 3888 \cos^{6} (x) - 1080 \cos^{4} (x) + 81 \cos^{2} (x)) - 1$

Step 1.2.8

Add $110592 \cos^{14} (x)$ and $331776 \cos^{14} (x)$ .

$1 + 2 (65536 \cos^{18} (x) - 294912 \cos^{16} (x) + 442368 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 110592 \cos^{14} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) + 69120 \cos^{10} (x) - 51840 \cos^{8} (x) + 14400 \cos^{6} (x) - 1080 \cos^{4} (x) + 2304 \cos^{10} (x) - 5184 \cos^{8} (x) + 3888 \cos^{6} (x) - 1080 \cos^{4} (x) + 81 \cos^{2} (x)) - 1$

Step 1.2.9

Add $442368 \cos^{14} (x)$ and $110592 \cos^{14} (x)$ .

$1 + 2 (65536 \cos^{18} (x) - 294912 \cos^{16} (x) + 552960 \cos^{14} (x) - 30720 \cos^{12} (x) + 2304 \cos^{10} (x) - 248832 \cos^{12} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) + 69120 \cos^{10} (x) - 51840 \cos^{8} (x) + 14400 \cos^{6} (x) - 1080 \cos^{4} (x) + 2304 \cos^{10} (x) - 5184 \cos^{8} (x) + 3888 \cos^{6} (x) - 1080 \cos^{4} (x) + 81 \cos^{2} (x)) - 1$

Step 1.2.10

Subtract $248832 \cos^{12} (x)$ from $- 30720 \cos^{12} (x)$ .

$1 + 2 (65536 \cos^{18} (x) - 294912 \cos^{16} (x) + 552960 \cos^{14} (x) - 279552 \cos^{12} (x) + 2304 \cos^{10} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) - 248832 \cos^{12} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) + 69120 \cos^{10} (x) - 51840 \cos^{8} (x) + 14400 \cos^{6} (x) - 1080 \cos^{4} (x) + 2304 \cos^{10} (x) - 5184 \cos^{8} (x) + 3888 \cos^{6} (x) - 1080 \cos^{4} (x) + 81 \cos^{2} (x)) - 1$

Step 1.2.11

Subtract $248832 \cos^{12} (x)$ from $- 279552 \cos^{12} (x)$ .

$1 + 2 (65536 \cos^{18} (x) - 294912 \cos^{16} (x) + 552960 \cos^{14} (x) - 528384 \cos^{12} (x) + 2304 \cos^{10} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 30720 \cos^{12} (x) + 69120 \cos^{10} (x) - 51840 \cos^{8} (x) + 14400 \cos^{6} (x) - 1080 \cos^{4} (x) + 2304 \cos^{10} (x) - 5184 \cos^{8} (x) + 3888 \cos^{6} (x) - 1080 \cos^{4} (x) + 81 \cos^{2} (x)) - 1$

Step 1.2.12

Subtract $30720 \cos^{12} (x)$ from $- 528384 \cos^{12} (x)$ .

$1 + 2 (65536 \cos^{18} (x) - 294912 \cos^{16} (x) + 552960 \cos^{14} (x) - 559104 \cos^{12} (x) + 2304 \cos^{10} (x) + 69120 \cos^{10} (x) - 5184 \cos^{8} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) + 69120 \cos^{10} (x) - 51840 \cos^{8} (x) + 14400 \cos^{6} (x) - 1080 \cos^{4} (x) + 2304 \cos^{10} (x) - 5184 \cos^{8} (x) + 3888 \cos^{6} (x) - 1080 \cos^{4} (x) + 81 \cos^{2} (x)) - 1$

Step 1.2.13

Add $2304 \cos^{10} (x)$ and $69120 \cos^{10} (x)$ .

$1 + 2 (65536 \cos^{18} (x) - 294912 \cos^{16} (x) + 552960 \cos^{14} (x) - 559104 \cos^{12} (x) + 71424 \cos^{10} (x) - 5184 \cos^{8} (x) + 186624 \cos^{10} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) + 69120 \cos^{10} (x) - 51840 \cos^{8} (x) + 14400 \cos^{6} (x) - 1080 \cos^{4} (x) + 2304 \cos^{10} (x) - 5184 \cos^{8} (x) + 3888 \cos^{6} (x) - 1080 \cos^{4} (x) + 81 \cos^{2} (x)) - 1$

Step 1.2.14

Add $71424 \cos^{10} (x)$ and $186624 \cos^{10} (x)$ .

$1 + 2 (65536 \cos^{18} (x) - 294912 \cos^{16} (x) + 552960 \cos^{14} (x) - 559104 \cos^{12} (x) + 258048 \cos^{10} (x) - 5184 \cos^{8} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) + 69120 \cos^{10} (x) - 51840 \cos^{8} (x) + 14400 \cos^{6} (x) - 1080 \cos^{4} (x) + 2304 \cos^{10} (x) - 5184 \cos^{8} (x) + 3888 \cos^{6} (x) - 1080 \cos^{4} (x) + 81 \cos^{2} (x)) - 1$

Step 1.2.15

Add $258048 \cos^{10} (x)$ and $69120 \cos^{10} (x)$ .

$1 + 2 (65536 \cos^{18} (x) - 294912 \cos^{16} (x) + 552960 \cos^{14} (x) - 559104 \cos^{12} (x) + 327168 \cos^{10} (x) - 5184 \cos^{8} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 51840 \cos^{8} (x) + 14400 \cos^{6} (x) - 1080 \cos^{4} (x) + 2304 \cos^{10} (x) - 5184 \cos^{8} (x) + 3888 \cos^{6} (x) - 1080 \cos^{4} (x) + 81 \cos^{2} (x)) - 1$

Step 1.2.16

Add $327168 \cos^{10} (x)$ and $2304 \cos^{10} (x)$ .

$1 + 2 (65536 \cos^{18} (x) - 294912 \cos^{16} (x) + 552960 \cos^{14} (x) - 559104 \cos^{12} (x) + 329472 \cos^{10} (x) - 5184 \cos^{8} (x) - 51840 \cos^{8} (x) + 3888 \cos^{6} (x) - 51840 \cos^{8} (x) + 14400 \cos^{6} (x) - 1080 \cos^{4} (x) - 5184 \cos^{8} (x) + 3888 \cos^{6} (x) - 1080 \cos^{4} (x) + 81 \cos^{2} (x)) - 1$

Step 1.2.17

Subtract $51840 \cos^{8} (x)$ from $- 5184 \cos^{8} (x)$ .

$1 + 2 (65536 \cos^{18} (x) - 294912 \cos^{16} (x) + 552960 \cos^{14} (x) - 559104 \cos^{12} (x) + 329472 \cos^{10} (x) - 57024 \cos^{8} (x) + 3888 \cos^{6} (x) - 51840 \cos^{8} (x) + 14400 \cos^{6} (x) - 1080 \cos^{4} (x) - 5184 \cos^{8} (x) + 3888 \cos^{6} (x) - 1080 \cos^{4} (x) + 81 \cos^{2} (x)) - 1$

Step 1.2.18

Subtract $51840 \cos^{8} (x)$ from $- 57024 \cos^{8} (x)$ .

$1 + 2 (65536 \cos^{18} (x) - 294912 \cos^{16} (x) + 552960 \cos^{14} (x) - 559104 \cos^{12} (x) + 329472 \cos^{10} (x) - 108864 \cos^{8} (x) + 3888 \cos^{6} (x) + 14400 \cos^{6} (x) - 1080 \cos^{4} (x) - 5184 \cos^{8} (x) + 3888 \cos^{6} (x) - 1080 \cos^{4} (x) + 81 \cos^{2} (x)) - 1$

Step 1.2.19

Subtract $5184 \cos^{8} (x)$ from $- 108864 \cos^{8} (x)$ .

$1 + 2 (65536 \cos^{18} (x) - 294912 \cos^{16} (x) + 552960 \cos^{14} (x) - 559104 \cos^{12} (x) + 329472 \cos^{10} (x) - 114048 \cos^{8} (x) + 3888 \cos^{6} (x) + 14400 \cos^{6} (x) - 1080 \cos^{4} (x) + 3888 \cos^{6} (x) - 1080 \cos^{4} (x) + 81 \cos^{2} (x)) - 1$

Step 1.2.20

Add $3888 \cos^{6} (x)$ and $14400 \cos^{6} (x)$ .

$1 + 2 (65536 \cos^{18} (x) - 294912 \cos^{16} (x) + 552960 \cos^{14} (x) - 559104 \cos^{12} (x) + 329472 \cos^{10} (x) - 114048 \cos^{8} (x) + 18288 \cos^{6} (x) - 1080 \cos^{4} (x) + 3888 \cos^{6} (x) - 1080 \cos^{4} (x) + 81 \cos^{2} (x)) - 1$

Step 1.2.21

Add $18288 \cos^{6} (x)$ and $3888 \cos^{6} (x)$ .

$1 + 2 (65536 \cos^{18} (x) - 294912 \cos^{16} (x) + 552960 \cos^{14} (x) - 559104 \cos^{12} (x) + 329472 \cos^{10} (x) - 114048 \cos^{8} (x) + 22176 \cos^{6} (x) - 1080 \cos^{4} (x) - 1080 \cos^{4} (x) + 81 \cos^{2} (x)) - 1$

Step 1.2.22

Subtract $1080 \cos^{4} (x)$ from $- 1080 \cos^{4} (x)$ .

$1 + 2 (65536 \cos^{18} (x) - 294912 \cos^{16} (x) + 552960 \cos^{14} (x) - 559104 \cos^{12} (x) + 329472 \cos^{10} (x) - 114048 \cos^{8} (x) + 22176 \cos^{6} (x) - 2160 \cos^{4} (x) + 81 \cos^{2} (x)) - 1$

Step 1.2.23

Apply the distributive property.

$1 + 2 (65536 \cos^{18} (x)) + 2 (- 294912 \cos^{16} (x)) + 2 (552960 \cos^{14} (x)) + 2 (- 559104 \cos^{12} (x)) + 2 (329472 \cos^{10} (x)) + 2 (- 114048 \cos^{8} (x)) + 2 (22176 \cos^{6} (x)) + 2 (- 2160 \cos^{4} (x)) + 2 (81 \cos^{2} (x)) - 1$

Step 1.2.24

Simplify.

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Step 1.2.24.1

Multiply $65536$ by $2$ .

$1 + 131072 \cos^{18} (x) + 2 (- 294912 \cos^{16} (x)) + 2 (552960 \cos^{14} (x)) + 2 (- 559104 \cos^{12} (x)) + 2 (329472 \cos^{10} (x)) + 2 (- 114048 \cos^{8} (x)) + 2 (22176 \cos^{6} (x)) + 2 (- 2160 \cos^{4} (x)) + 2 (81 \cos^{2} (x)) - 1$

Step 1.2.24.2

Multiply $- 294912$ by $2$ .

$1 + 131072 \cos^{18} (x) - 589824 \cos^{16} (x) + 2 (552960 \cos^{14} (x)) + 2 (- 559104 \cos^{12} (x)) + 2 (329472 \cos^{10} (x)) + 2 (- 114048 \cos^{8} (x)) + 2 (22176 \cos^{6} (x)) + 2 (- 2160 \cos^{4} (x)) + 2 (81 \cos^{2} (x)) - 1$

Step 1.2.24.3

Multiply $552960$ by $2$ .

$1 + 131072 \cos^{18} (x) - 589824 \cos^{16} (x) + 1105920 \cos^{14} (x) + 2 (- 559104 \cos^{12} (x)) + 2 (329472 \cos^{10} (x)) + 2 (- 114048 \cos^{8} (x)) + 2 (22176 \cos^{6} (x)) + 2 (- 2160 \cos^{4} (x)) + 2 (81 \cos^{2} (x)) - 1$

Step 1.2.24.4

Multiply $- 559104$ by $2$ .

$1 + 131072 \cos^{18} (x) - 589824 \cos^{16} (x) + 1105920 \cos^{14} (x) - 1118208 \cos^{12} (x) + 2 (329472 \cos^{10} (x)) + 2 (- 114048 \cos^{8} (x)) + 2 (22176 \cos^{6} (x)) + 2 (- 2160 \cos^{4} (x)) + 2 (81 \cos^{2} (x)) - 1$

Step 1.2.24.5

Multiply $329472$ by $2$ .

$1 + 131072 \cos^{18} (x) - 589824 \cos^{16} (x) + 1105920 \cos^{14} (x) - 1118208 \cos^{12} (x) + 658944 \cos^{10} (x) + 2 (- 114048 \cos^{8} (x)) + 2 (22176 \cos^{6} (x)) + 2 (- 2160 \cos^{4} (x)) + 2 (81 \cos^{2} (x)) - 1$

Step 1.2.24.6

Multiply $- 114048$ by $2$ .

$1 + 131072 \cos^{18} (x) - 589824 \cos^{16} (x) + 1105920 \cos^{14} (x) - 1118208 \cos^{12} (x) + 658944 \cos^{10} (x) - 228096 \cos^{8} (x) + 2 (22176 \cos^{6} (x)) + 2 (- 2160 \cos^{4} (x)) + 2 (81 \cos^{2} (x)) - 1$

Step 1.2.24.7

Multiply $22176$ by $2$ .

$1 + 131072 \cos^{18} (x) - 589824 \cos^{16} (x) + 1105920 \cos^{14} (x) - 1118208 \cos^{12} (x) + 658944 \cos^{10} (x) - 228096 \cos^{8} (x) + 44352 \cos^{6} (x) + 2 (- 2160 \cos^{4} (x)) + 2 (81 \cos^{2} (x)) - 1$

Step 1.2.24.8

Multiply $- 2160$ by $2$ .

$1 + 131072 \cos^{18} (x) - 589824 \cos^{16} (x) + 1105920 \cos^{14} (x) - 1118208 \cos^{12} (x) + 658944 \cos^{10} (x) - 228096 \cos^{8} (x) + 44352 \cos^{6} (x) - 4320 \cos^{4} (x) + 2 (81 \cos^{2} (x)) - 1$

Step 1.2.24.9

Multiply $81$ by $2$ .

Step 2

Combine the opposite terms in $1 + 131072 \cos^{18} (x) - 589824 \cos^{16} (x) + 1105920 \cos^{14} (x) - 1118208 \cos^{12} (x) + 658944 \cos^{10} (x) - 228096 \cos^{8} (x) + 44352 \cos^{6} (x) - 4320 \cos^{4} (x) + 162 \cos^{2} (x) - 1$ .

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Step 2.1

Subtract $1$ from $1$ .

$0 + 131072 \cos^{18} (x) - 589824 \cos^{16} (x) + 1105920 \cos^{14} (x) - 1118208 \cos^{12} (x) + 658944 \cos^{10} (x) - 228096 \cos^{8} (x) + 44352 \cos^{6} (x) - 4320 \cos^{4} (x) + 162 \cos^{2} (x)$

Step 2.2

Add $0$ and $131072 \cos^{18} (x)$ .

$131072 \cos^{18} (x) - 589824 \cos^{16} (x) + 1105920 \cos^{14} (x) - 1118208 \cos^{12} (x) + 658944 \cos^{10} (x) - 228096 \cos^{8} (x) + 44352 \cos^{6} (x) - 4320 \cos^{4} (x) + 162 \cos^{2} (x)$