Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4

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

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

Step 5

Separate fractions.

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

Step 6

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

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

Step 7

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\frac{1}{\cos (x) - \sec (x)} \tan (x)$

Step 8

Combine $\frac{1}{\cos (x) - \sec (x)}$ and $\tan (x)$ .

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