Simplify sin(x)^2(sec(x)^2+csc(x)^2)

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

Step 1

Simplify each term.

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

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

Apply the product rule to $\frac{1}{\cos (x)}$ .

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

One to any power is one.

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

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

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

Apply the product rule to $\frac{1}{\sin (x)}$ .

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

One to any power is one.

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

Step 2

Simplify terms.

Apply the distributive property.

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

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

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

Cancel the common factor of $\sin^{2} (x)$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 3

Convert from $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ to $\tan^{2} (x)$ .

$\tan^{2} (x) + 1$

Step 4

Apply pythagorean identity.

$\sec^{2} (x)$