Precalculus Examples

$\frac{1 + \tan^{2} (x)}{\tan^{2} (x)} = \csc^{2} (x)$

Step 1

Start on the left side.

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

Step 2

Apply Pythagorean identity.

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Step 2.1

Rearrange terms.

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

Step 2.2

Apply pythagorean identity.

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

Step 3

Convert to sines and cosines.

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Step 3.1

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

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

Step 3.2

Write $\tan (x)$ in sines and cosines using the quotient identity.

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

Step 3.3

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

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

Step 3.4

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

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

Step 4

Simplify.

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.2

Combine.

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

Step 4.3

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

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Step 4.3.1

Cancel the common factor.

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

Step 4.3.2

Rewrite the expression.

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

Step 4.4

One to any power is one.

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

Step 5

Rewrite $\frac{1}{\sin^{2} (x)}$ as $\csc^{2} (x)$ .

$\csc^{2} (x)$

Step 6

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

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