Simplify cot(x)(1/(sec(x))+1/(csc(x)))

$\cot (x) (\frac{1}{\sec (x)} + \frac{1}{\csc (x)})$

Simplify each term.

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

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

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (x)}$ .

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

Multiply $\cos (x)$ by $1$ .

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

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

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

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\sin (x)}$ .

$\cot (x) (\cos (x) + 1 \sin (x))$

Multiply $\sin (x)$ by $1$ .

$\cot (x) (\cos (x) + \sin (x))$

Apply the distributive property.

$\cot (x) \cos (x) + \cot (x) \sin (x)$

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

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

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

Cancel the common factors.

$\cot (x) \cos (x) + \cos (x)$