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

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

Simplify each term.

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

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

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

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

Write $\cos (- x)$ as a fraction with denominator $1$ .

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

Simplify.

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

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

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

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

Simplify the numerator.

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 $- \sin (x)$ .

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

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

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

Move the negative in front of the fraction.

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

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

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

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 $1$ .

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