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