Simplify (csc(theta)*cos(theta))/(tan(theta)+cot(theta))

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

Step 1

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

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

Step 2

Simplify the denominator.

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

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

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

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

Step 3

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

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

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

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

Step 6

Separate fractions.

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

Step 7

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

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

Step 8

Simplify each term.

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

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

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

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

Step 9

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

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