Algebra Examples

\frac{\tan (t)}{\sec (t) - \cos (t)}

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

Step 1

Rewrite

\tan (t)

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

\frac{\frac{\sin (t)}{\cos (t)}}{\sec (t) - \cos (t)}

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

Step 2

Rewrite

\sec (t)

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

\frac{\frac{\sin (t)}{\cos (t)}}{\frac{1}{\cos (t)} - \cos (t)}

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

\frac{\sin (t)}{\cos (t)} \cdot \frac{1}{\frac{1}{\cos (t)} - \cos (t)}

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

Step 4

Multiply

\frac{\sin (t)}{\cos (t)}

$\frac{\sin (t)}{\cos (t)}$ by

\frac{1}{\frac{1}{\cos (t)} - \cos (t)}

$\frac{1}{\frac{1}{\cos (t)} - \cos (t)}$ .

\frac{\sin (t)}{\cos (t) (\frac{1}{\cos (t)} - \cos (t))}

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

Step 5

Separate fractions.

\frac{1}{\frac{1}{\cos (t)} - \cos (t)} \cdot \frac{\sin (t)}{\cos (t)}

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

Step 6

Convert from

\frac{\sin (t)}{\cos (t)}

$\frac{\sin (t)}{\cos (t)}$ to

\tan (t)

$\tan (t)$ .

\frac{1}{\frac{1}{\cos (t)} - \cos (t)} \tan (t)

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

Step 7

Convert from

\frac{1}{\cos (t)}

$\frac{1}{\cos (t)}$ to

\sec (t)

$\sec (t)$ .

\frac{1}{\sec (t) - \cos (t)} \tan (t)

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

Step 8

Combine

\frac{1}{\sec (t) - \cos (t)}

$\frac{1}{\sec (t) - \cos (t)}$ and

\tan (t)

$\tan (t)$ .

\frac{\tan (t)}{\sec (t) - \cos (t)}

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