Trigonometry Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Multiply by the reciprocal of the fraction to divide by $\frac{\cos (- x)}{\sin (- x)}$ .

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

Step 4

Simplify.

Tap for more steps...

Step 4.1

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

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

Step 4.2

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

$\csc (x) \tan (- x)$

Step 5

Since $\tan (- x)$ is an odd function, rewrite $\tan (- x)$ as $- \tan (x)$ .

$\csc (x) (- \tan (x))$

Step 6

Rewrite in terms of sines and cosines, then cancel the common factors.

Tap for more steps...

Step 6.1

Reorder $\csc (x)$ and $- \tan (x)$ .

$- \tan (x) \csc (x)$

Step 6.2

Add parentheses.

$- (\tan (x) \csc (x))$

Step 6.3

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

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

Step 6.4

Cancel the common factors.

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

Step 7

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

$- \sec (x)$