Trigonometry Examples

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

Step 1

Simplify $(\cos (x) + \sin (x)) (1 - \cos (x) \sin (x))$ .

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Step 1.1

Expand $(\cos (x) + \sin (x)) (1 - \cos (x) \sin (x))$ using the FOIL Method.

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Step 1.1.1

Apply the distributive property.

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

Step 1.1.2

Apply the distributive property.

$\cos (x) \cdot 1 + \cos (x) (- \cos (x) \sin (x)) + \sin (x) (1 - \cos (x) \sin (x)) = \sin^{3} (x) + \cos^{3} (x)$

Step 1.1.3

Apply the distributive property.

$\cos (x) \cdot 1 + \cos (x) (- \cos (x) \sin (x)) + \sin (x) \cdot 1 + \sin (x) (- \cos (x) \sin (x)) = \sin^{3} (x) + \cos^{3} (x)$

Step 1.2

Simplify each term.

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Step 1.2.1

Multiply $\cos (x)$ by $1$ .

$\cos (x) + \cos (x) (- \cos (x) \sin (x)) + \sin (x) \cdot 1 + \sin (x) (- \cos (x) \sin (x)) = \sin^{3} (x) + \cos^{3} (x)$

Step 1.2.2

Rewrite using the commutative property of multiplication.

$\cos (x) - \cos (x) \cos (x) \sin (x) + \sin (x) \cdot 1 + \sin (x) (- \cos (x) \sin (x)) = \sin^{3} (x) + \cos^{3} (x)$

Step 1.2.3

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

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Step 1.2.3.1

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

$\cos (x) - (\cos^{1} (x) \cos (x)) \sin (x) + \sin (x) \cdot 1 + \sin (x) (- \cos (x) \sin (x)) = \sin^{3} (x) + \cos^{3} (x)$

Step 1.2.3.2

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

$\cos (x) - (\cos^{1} (x) \cos^{1} (x)) \sin (x) + \sin (x) \cdot 1 + \sin (x) (- \cos (x) \sin (x)) = \sin^{3} (x) + \cos^{3} (x)$

Step 1.2.3.3

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

$\cos (x) - {\cos (x)}^{1 + 1} \sin (x) + \sin (x) \cdot 1 + \sin (x) (- \cos (x) \sin (x)) = \sin^{3} (x) + \cos^{3} (x)$

Step 1.2.3.4

Add $1$ and $1$ .

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

Step 1.2.4

Multiply $\sin (x)$ by $1$ .

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

Step 1.2.5

Rewrite using the commutative property of multiplication.

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

Step 1.2.6

Multiply $- \sin (x) \cos (x) \sin (x)$ .

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Step 1.2.6.1

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

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

Step 1.2.6.2

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

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

Step 1.2.6.3

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

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

Step 1.2.6.4

Add $1$ and $1$ .

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

Step 2

Simplify $\sin^{3} (x) + \cos^{3} (x)$ .

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Step 2.1

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = \sin (x)$ and $b = \cos (x)$ .

$\cos (x) - \cos^{2} (x) \sin (x) + \sin (x) - \sin^{2} (x) \cos (x) = (\sin (x) + \cos (x)) (\sin^{2} (x) - \sin (x) \cos (x) + \cos^{2} (x))$

Step 2.2

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

$\cos (x) - \cos^{2} (x) \sin (x) + \sin (x) - \sin^{2} (x) \cos (x) = (\sin (x) + \cos (x)) (\sin^{2} (x) + \cos^{2} (x) - \sin (x) \cos (x))$

Step 2.3

Apply pythagorean identity.

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

Step 2.4

Expand $(\sin (x) + \cos (x)) (1 - \sin (x) \cos (x))$ using the FOIL Method.

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Step 2.4.1

Apply the distributive property.

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

Step 2.4.2

Apply the distributive property.

$\cos (x) - \cos^{2} (x) \sin (x) + \sin (x) - \sin^{2} (x) \cos (x) = \sin (x) \cdot 1 + \sin (x) (- \sin (x) \cos (x)) + \cos (x) (1 - \sin (x) \cos (x))$

Step 2.4.3

Apply the distributive property.

$\cos (x) - \cos^{2} (x) \sin (x) + \sin (x) - \sin^{2} (x) \cos (x) = \sin (x) \cdot 1 + \sin (x) (- \sin (x) \cos (x)) + \cos (x) \cdot 1 + \cos (x) (- \sin (x) \cos (x))$

Step 2.5

Simplify each term.

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Step 2.5.1

Multiply $\sin (x)$ by $1$ .

$\cos (x) - \cos^{2} (x) \sin (x) + \sin (x) - \sin^{2} (x) \cos (x) = \sin (x) + \sin (x) (- \sin (x) \cos (x)) + \cos (x) \cdot 1 + \cos (x) (- \sin (x) \cos (x))$

Step 2.5.2

Rewrite using the commutative property of multiplication.

$\cos (x) - \cos^{2} (x) \sin (x) + \sin (x) - \sin^{2} (x) \cos (x) = \sin (x) - \sin (x) \sin (x) \cos (x) + \cos (x) \cdot 1 + \cos (x) (- \sin (x) \cos (x))$

Step 2.5.3

Multiply $- \sin (x) \sin (x)$ .

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Step 2.5.3.1

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

$\cos (x) - \cos^{2} (x) \sin (x) + \sin (x) - \sin^{2} (x) \cos (x) = \sin (x) - (\sin^{1} (x) \sin (x)) \cos (x) + \cos (x) \cdot 1 + \cos (x) (- \sin (x) \cos (x))$

Step 2.5.3.2

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

$\cos (x) - \cos^{2} (x) \sin (x) + \sin (x) - \sin^{2} (x) \cos (x) = \sin (x) - (\sin^{1} (x) \sin^{1} (x)) \cos (x) + \cos (x) \cdot 1 + \cos (x) (- \sin (x) \cos (x))$

Step 2.5.3.3

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

$\cos (x) - \cos^{2} (x) \sin (x) + \sin (x) - \sin^{2} (x) \cos (x) = \sin (x) - {\sin (x)}^{1 + 1} \cos (x) + \cos (x) \cdot 1 + \cos (x) (- \sin (x) \cos (x))$

Step 2.5.3.4

Add $1$ and $1$ .

$\cos (x) - \cos^{2} (x) \sin (x) + \sin (x) - \sin^{2} (x) \cos (x) = \sin (x) - \sin^{2} (x) \cos (x) + \cos (x) \cdot 1 + \cos (x) (- \sin (x) \cos (x))$

Step 2.5.4

Multiply $\cos (x)$ by $1$ .

$\cos (x) - \cos^{2} (x) \sin (x) + \sin (x) - \sin^{2} (x) \cos (x) = \sin (x) - \sin^{2} (x) \cos (x) + \cos (x) + \cos (x) (- \sin (x) \cos (x))$

Step 2.5.5

Rewrite using the commutative property of multiplication.

$\cos (x) - \cos^{2} (x) \sin (x) + \sin (x) - \sin^{2} (x) \cos (x) = \sin (x) - \sin^{2} (x) \cos (x) + \cos (x) - \cos (x) \sin (x) \cos (x)$

Step 2.5.6

Multiply $- \cos (x) \sin (x) \cos (x)$ .

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Step 2.5.6.1

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

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

Step 2.5.6.2

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

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

Step 2.5.6.3

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

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

Step 2.5.6.4

Add $1$ and $1$ .

$\cos (x) - \cos^{2} (x) \sin (x) + \sin (x) - \sin^{2} (x) \cos (x) = \sin (x) - \sin^{2} (x) \cos (x) + \cos (x) - \cos^{2} (x) \sin (x)$

Step 3

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx 0, - 0.1, 0.1, \frac{1}{5}, - 0.3, 0.3, - \frac{2}{5}$

Step 4