Precalculus Examples

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

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

Step 1

Simplify with factoring out.

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Step 1.1

Reorder

{cos}^{2} (x)

$\cos^{2} (x)$ and

- 1

$- 1$ .

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

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

Step 1.2

Rewrite

- 1

$- 1$ as

- 1 (1)

$- 1 (1)$ .

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

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

Step 1.3

Factor

- 1

$- 1$ out of

{cos}^{2} (x)

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

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

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

Step 1.4

Factor

- 1

$- 1$ out of

- 1 (1) - 1 (- {cos}^{2} (x))

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

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

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

Step 1.5

Rewrite

- 1 (1 - {cos}^{2} (x))

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

- (1 - {cos}^{2} (x))

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

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

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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