Trigonometry Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4

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

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

Step 5

Separate fractions.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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