Simplify cos(x)(cot(x)+tan(x))

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

Simplify each term.

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

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

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

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

Apply the distributive property.

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

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

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Cancel the common factor of $\cos (x)$ .

Cancel the common factor.

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

Rewrite the expression.

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

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 $1$ .

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