Simplify sec(x)csc(x)(1-sin(x)^2)

$\sec (x) \csc (x) (1 - \sin^{2} (x))$

Step 1

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

$\frac{1}{\cos (x)} \csc (x) (1 - \sin^{2} (x))$

Step 2

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

$\frac{1}{\cos (x)} \cdot \frac{1}{\sin (x)} (1 - \sin^{2} (x))$

Step 3

Multiply $\frac{1}{\cos (x)}$ by $\frac{1}{\sin (x)}$ .

$\frac{1}{\cos (x) \sin (x)} (1 - \sin^{2} (x))$

Step 4

Apply pythagorean identity.

$\frac{1}{\cos (x) \sin (x)} \cos^{2} (x)$

Step 5

Simplify terms.

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

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

$\frac{1}{\cos (x) (\sin (x))} \cos^{2} (x)$

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Step 6

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

$\cot (x)$