Evaluate cot(-x)cos(-x)+sin(-x)

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

Step 1

Simplify each term.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Step 2

Simplify each term.

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

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

Separate fractions.

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

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

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

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

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