Simplify/Condense (tan(t))/(sec(t)-cos(t))

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

Step 1

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

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

Step 2

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

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4

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

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

Step 5

Separate fractions.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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