Simplify (cot(x)^2)/(1+cot(x)^2)

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

Apply pythagorean identity.

$\frac{\cot^{2} (x)}{\csc^{2} (x)}$

Rewrite $\frac{\cot^{2} (x)}{\csc^{2} (x)}$ as ${(\frac{\cot (x)}{\csc (x)})}^{2}$ .

${(\frac{\cot (x)}{\csc (x)})}^{2}$

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

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

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

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

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\sin (x)}$ .

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

Write $\sin (x)$ as a fraction with denominator $1$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

$\cos^{2} (x)$