Precalculus Examples

$\frac{\sec (x)}{\tan (x)} = \frac{\tan (x)}{\sec (x) - \cos (x)}$

Step 1

Start on the right side.

$\frac{\tan (x)}{\sec (x) - \cos (x)}$

Step 2

Simplify the expression.

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Step 2.1

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

$\frac{\frac{\sin (x)}{\cos (x)}}{\sec (x) - \cos (x)}$

Step 2.2

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

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

Step 2.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.4

Multiply $\frac{\sin (x)}{\cos (x)}$ by $\frac{1}{\frac{1}{\cos (x)} - \cos (x)}$ .

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

Step 3

Apply the distributive property.

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

Step 4

Simplify each term.

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

Step 5

Apply pythagorean identity.

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

Step 6

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

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

Step 7

Rewrite $\frac{1}{\sin (x)}$ as $\frac{\sec (x)}{\tan (x)}$ .

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

Step 8

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

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