Precalculus Examples

$\frac{\cot (t)}{\csc (t) - \sin (t)}$

Step 1

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

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

Step 2

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

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4

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

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

Step 5

Separate fractions.

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

Step 6

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

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

Step 7

Convert from $\frac{1}{\sin (t)}$ to $\csc (t)$ .

$\frac{1}{\csc (t) - \sin (t)} \cot (t)$

Step 8

Combine $\frac{1}{\csc (t) - \sin (t)}$ and $\cot (t)$ .

$\frac{\cot (t)}{\csc (t) - \sin (t)}$