Simplify sin(x)(sec(x)-csc(x))

$\sin (x) (\sec (x) - \csc (x))$

Step 1

Simplify each term.

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

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

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

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

Step 2

Simplify terms.

Apply the distributive property.

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

Combine $\sin (x)$ and $\frac{1}{\cos (x)}$ .

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

Rewrite using the commutative property of multiplication.

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

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

Factor $\sin (x)$ out of $- \sin (x)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 3

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

$\tan (x) - 1$