Trigonometry Examples

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

Step 1

Simplify with factoring out.

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Step 1.1

Reorder $\cos^{2} (x)$ and $- 1$ .

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

Step 1.2

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 1.3

Factor $- 1$ out of $\cos^{2} (x)$ .

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

Step 1.4

Factor $- 1$ out of $- 1 (1) - 1 (- \cos^{2} (x))$ .

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

Step 1.5

Rewrite $- 1 (1 - \cos^{2} (x))$ as $- (1 - \cos^{2} (x))$ .

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

Step 2

Apply pythagorean identity.

$- \sin^{2} (x) (\tan^{2} (x) - 1)$

Step 3

Simplify terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Rewrite $\tan (x)$ in terms of sines and cosines.

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

Step 3.1.2

Apply the product rule to $\frac{\sin (x)}{\cos (x)}$ .

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

Step 3.2

Apply the distributive property.

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

Step 4

Multiply $- \sin^{2} (x) \frac{\sin^{2} (x)}{\cos^{2} (x)}$ .

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Step 4.1

Combine $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ and $\sin^{2} (x)$ .

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

Add $2$ and $2$ .

$- \frac{\sin^{4} (x)}{\cos^{2} (x)} - \sin^{2} (x) \cdot - 1$

Step 5

Multiply $- \sin^{2} (x) \cdot - 1$ .

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Step 5.1

Multiply $- 1$ by $- 1$ .

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

Step 5.2

Multiply $\sin^{2} (x)$ by $1$ .

$- \frac{\sin^{4} (x)}{\cos^{2} (x)} + \sin^{2} (x)$

Step 6

Simplify the expression.

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Step 6.1

Rewrite $\frac{\sin^{4} (x)}{\cos^{2} (x)}$ as ${(\frac{\sin^{2} (x)}{\cos (x)})}^{2}$ .

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

Step 6.2

Reorder $- {(\frac{\sin^{2} (x)}{\cos (x)})}^{2}$ and $\sin^{2} (x)$ .

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

Step 7

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

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

Step 8

Simplify terms.

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Step 8.1

Simplify each term.

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Step 8.1.1

Factor $\sin (x)$ out of $\sin^{2} (x)$ .

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

Step 8.1.2

Separate fractions.

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

Step 8.1.3

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

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

Step 8.1.4

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

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

Step 8.2

Simplify each term.

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Step 8.2.1

Factor $\sin (x)$ out of $\sin^{2} (x)$ .

$(\sin (x) + \sin (x) \tan (x)) (\sin (x) - \frac{\sin (x) \sin (x)}{\cos (x)})$

Step 8.2.2

Separate fractions.

$(\sin (x) + \sin (x) \tan (x)) (\sin (x) - (\frac{\sin (x)}{1} \cdot \frac{\sin (x)}{\cos (x)}))$

Step 8.2.3

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$(\sin (x) + \sin (x) \tan (x)) (\sin (x) - (\frac{\sin (x)}{1} \tan (x)))$

Step 8.2.4

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

$(\sin (x) + \sin (x) \tan (x)) (\sin (x) - \sin (x) \tan (x))$

Step 9

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

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Step 9.1

Apply the distributive property.

$\sin (x) (\sin (x) - \sin (x) \tan (x)) + \sin (x) \tan (x) (\sin (x) - \sin (x) \tan (x))$

Step 9.2

Apply the distributive property.

$\sin (x) \sin (x) + \sin (x) (- \sin (x) \tan (x)) + \sin (x) \tan (x) (\sin (x) - \sin (x) \tan (x))$

Step 9.3

Apply the distributive property.

$\sin (x) \sin (x) + \sin (x) (- \sin (x) \tan (x)) + \sin (x) \tan (x) \sin (x) + \sin (x) \tan (x) (- \sin (x) \tan (x))$

Step 10

Simplify terms.

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Step 10.1

Combine the opposite terms in $\sin (x) \sin (x) + \sin (x) (- \sin (x) \tan (x)) + \sin (x) \tan (x) \sin (x) + \sin (x) \tan (x) (- \sin (x) \tan (x))$ .

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Step 10.1.1

Reorder the factors in the terms $\sin (x) (- \sin (x) \tan (x))$ and $\sin (x) \tan (x) \sin (x)$ .

$\sin (x) \sin (x) - \sin (x) \sin (x) \tan (x) + \sin (x) \sin (x) \tan (x) + \sin (x) \tan (x) (- \sin (x) \tan (x))$

Step 10.1.2

Add $- \sin (x) \sin (x) \tan (x)$ and $\sin (x) \sin (x) \tan (x)$ .

$\sin (x) \sin (x) + 0 + \sin (x) \tan (x) (- \sin (x) \tan (x))$

Step 10.1.3

Add $\sin (x) \sin (x)$ and $0$ .

$\sin (x) \sin (x) + \sin (x) \tan (x) (- \sin (x) \tan (x))$

Step 10.2

Simplify each term.

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Step 10.2.1

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

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Step 10.2.1.1

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

$\sin^{1} (x) \sin (x) + \sin (x) \tan (x) (- \sin (x) \tan (x))$

Step 10.2.1.2

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

$\sin^{1} (x) \sin^{1} (x) + \sin (x) \tan (x) (- \sin (x) \tan (x))$

Step 10.2.1.3

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

${\sin (x)}^{1 + 1} + \sin (x) \tan (x) (- \sin (x) \tan (x))$

Step 10.2.1.4

Add $1$ and $1$ .

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

Step 10.2.2

Rewrite using the commutative property of multiplication.

$\sin^{2} (x) - \sin (x) \tan (x) \sin (x) \tan (x)$

Step 10.2.3

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

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Step 10.2.3.1

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

$\sin^{2} (x) - (\sin^{1} (x) \sin (x)) \tan (x) \tan (x)$

Step 10.2.3.2

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

$\sin^{2} (x) - (\sin^{1} (x) \sin^{1} (x)) \tan (x) \tan (x)$

Step 10.2.3.3

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

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

Step 10.2.3.4

Add $1$ and $1$ .

$\sin^{2} (x) - \sin^{2} (x) \tan (x) \tan (x)$

Step 10.2.4

Multiply $- \sin^{2} (x) \tan (x) \tan (x)$ .

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Step 10.2.4.1

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

$\sin^{2} (x) - \sin^{2} (x) (\tan^{1} (x) \tan (x))$

Step 10.2.4.2

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

$\sin^{2} (x) - \sin^{2} (x) (\tan^{1} (x) \tan^{1} (x))$

Step 10.2.4.3

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

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

Step 10.2.4.4

Add $1$ and $1$ .

$\sin^{2} (x) - \sin^{2} (x) \tan^{2} (x)$

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