Precalculus Examples

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

Step 1

Start on the left side.

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

Step 2

Apply Pythagorean identity in reverse.

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

Step 3

Convert to sines and cosines.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 4

Simplify.

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Step 4.1

Subtract $1$ from $2$ .

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

Step 4.2

Simplify each term.

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Step 4.2.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.2.2

One to any power is one.

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

Step 4.2.3

One to any power is one.

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

Step 4.2.4

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

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

Step 4.2.5

Apply the distributive property.

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

Step 4.2.6

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

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

Step 4.2.7

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

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Step 4.2.7.1

Cancel the common factor.

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

Step 4.2.7.2

Rewrite the expression.

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

Step 4.3

Subtract $1$ from $1$ .

${\cos (x)}^{2} + 0$

Step 4.4

Add ${\cos (x)}^{2}$ and $0$ .

$\cos^{2} (x)$

Step 5

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

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