Simplify (cot(x)^2+1)(sec(x)^2-1)

$(\cot^{2} (x) + 1) (\sec^{2} (x) - 1)$

Step 1

Rearrange terms.

$(1 + \cot^{2} (x)) (\sec^{2} (x) - 1)$

Step 2

Apply pythagorean identity.

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

Step 3

Apply pythagorean identity.

$\csc^{2} (x) \tan^{2} (x)$

Step 4

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

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

Step 5

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

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

Step 6

Simplify the expression.

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

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

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

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

One to any power is one.

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

Step 7

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

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

Step 8

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 9

Rewrite $1$ as $1^{2}$ .

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

Step 10

Rewrite $\frac{1^{2}}{\cos^{2} (x)}$ as ${(\frac{1}{\cos (x)})}^{2}$ .

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

Step 11

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\sec^{2} (x)$