Answer
Work on the left side of the identity using $\cot{\theta}=\frac{\cos\theta}{\sin\theta}$ and $1-\cos^2\theta=\sin^2\theta$.
Refer to the step-by-step part below for the complete proof.
Work Step by Step
We have to show that:
$(1-\cos^2\theta)(1+\cot^2\theta)=1$
Since $1-\cos^2\theta=\sin^2\theta$, then the left side of the identity is equivalent to:
$=\sin^2\theta(1+\cot^2\theta)$
Distribute $\sin^2\theta$, then use $\cot\theta=\frac{\cos\theta}{\sin\theta}$ and $\sin^2\theta+\cos^2\theta=1$ to obtain:
$=\sin^2\theta+\sin^2\theta\cot^2\theta$
$=\sin^2\theta+\sin^2\theta\left(\frac{\cos^2\theta}{\sin^2\theta}\right)$
$=\sin^2\theta+\cos^2\theta=1$
Since LHS=RHS, then the identity's proof is complete.