Trigonometry Examples

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

Step 1

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

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

Step 2

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

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

Step 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 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 5

Separate fractions.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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