Precalculus Examples

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

Step 1

Start on the left side.

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

Step 2

Apply Pythagorean identity.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Apply pythagorean identity.

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

Step 2.7

Apply pythagorean identity.

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

Step 3

Apply Pythagorean identity in reverse.

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

Step 4

Convert to sines and cosines.

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Step 4.1

Apply the reciprocal identity to $\sec (x)$ .

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

Step 4.2

Apply the product rule to $\frac{1}{\cos (x)}$ .

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

Step 5

Simplify.

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Step 5.1

One to any power is one.

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Multiply $- 1$ by $1$ .

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

Step 5.4

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

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Step 5.4.1

Multiply $- 1$ by $- 1$ .

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

Step 5.4.2

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

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

Step 5.5

Apply the distributive property.

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

Step 5.6

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

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

Step 5.7

Cancel the common factor of ${\cos (x)}^{2}$ .

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

Step 6

Reorder terms.

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

Step 7

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

$1 - \sec^{2} (x)$

Step 8

Because the two sides have been shown to be equivalent, the equation is an identity.

$(\cos^{2} (x) - 1) (\tan^{2} (x) + 1) = 1 - \sec^{2} (x)$ is an identity