Simplify/Condense cos(-x)+(sin(-x))/(cot(-x))

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

Simplify each term.

Since $\cos (- x)$ is an even function, rewrite $\cos (- x)$ as $\cos (x)$ .

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

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

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

Simplify the denominator.

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

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

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

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

Dividing two negative values results in a positive value.

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

Multiply the numerator by the reciprocal of the denominator.

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

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

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

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

Raise $\sin (x)$ to the power of $1$ .

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

Raise $\sin (x)$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $1$ and $1$ .

$\cos (x) + \frac{\sin^{2} (x)}{\cos (x)}$

Simplify each term.

Factor $\sin (x)$ out of $\sin^{2} (x)$ .

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

Separate fractions.

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

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

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

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

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