Simplify sin(x)*(tan(x)+cot(x))

$\sin (x) \cdot (\tan (x) + \cot (x))$

Step 1

Simplify each term.

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

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

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

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

Step 2

Apply the distributive property.

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

Step 3

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

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Step 4

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 5

Simplify each term.

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

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

Separate fractions.

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

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

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

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

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