Evaluate (csc(x)cos(x))/(tan(x)+cot(x))

$\frac{\csc (x) \cos (x)}{\tan (x) + \cot (x)}$

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

$\frac{\frac{1}{\sin (x)} \cos (x)}{\tan (x) + \cot (x)}$

Simplify the denominator.

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

$\frac{\frac{1}{\sin (x)} \cos (x)}{\frac{\sin (x)}{\cos (x)} + \cot (x)}$

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

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

Combine $\frac{1}{\sin (x)}$ and $\cos (x)$ .

$\frac{\frac{\cos (x)}{\sin (x)}}{\frac{\sin (x)}{\cos (x)} + \frac{\cos (x)}{\sin (x)}}$

Multiply the numerator by the reciprocal of the denominator.

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

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

$\frac{\cos (x)}{\sin (x) (\frac{\sin (x)}{\cos (x)} + \frac{\cos (x)}{\sin (x)})}$

Separate fractions.

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

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

$\frac{1}{\frac{\sin (x)}{\cos (x)} + \frac{\cos (x)}{\sin (x)}} \cot (x)$

Simplify each term.

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

$\frac{1}{\tan (x) + \frac{\cos (x)}{\sin (x)}} \cot (x)$

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

$\frac{1}{\tan (x) + \cot (x)} \cot (x)$

Combine $\frac{1}{\tan (x) + \cot (x)}$ and $\cot (x)$ .

$\frac{\cot (x)}{\tan (x) + \cot (x)}$