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

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

Step 1

Simplify terms.

Simplify each term.

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

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

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

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

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

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

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

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

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

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

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

Multiply $\cos^{2} (x)$ by $\cos^{2} (x)$ by adding the exponents.

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

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

Add $2$ and $2$ .

$1 + \frac{\cos^{2} (x)}{\sin^{2} (x)} - \cos^{2} (x) - \frac{\cos^{4} (x)}{\sin^{2} (x)}$

Move $- \cos^{2} (x)$ .

$1 - \cos^{2} (x) + \frac{\cos^{2} (x)}{\sin^{2} (x)} - \frac{\cos^{4} (x)}{\sin^{2} (x)}$

Step 2

Apply pythagorean identity.

$\sin^{2} (x) + \frac{\cos^{2} (x)}{\sin^{2} (x)} - \frac{\cos^{4} (x)}{\sin^{2} (x)}$

Step 3

Simplify each term.

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

$\sin^{2} (x) + \cot^{2} (x) - \frac{\cos^{4} (x)}{\sin^{2} (x)}$

Factor $\cos^{2} (x)$ out of $\cos^{4} (x)$ .

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

Multiply by $1$ .

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

Separate fractions.

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

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

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

Divide $\cos^{2} (x)$ by $1$ .

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