Answer
Work on the left side of the identity using $\tan{\theta}=\frac{\sin\theta}{\cos\theta}$ 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:
$\cos^2\theta(1+\tan^2\theta)=1$
By evaluating the left side we get:
$\cos^2\theta(1+\tan^2\theta)\\
=\cos^2\theta+\cos^2\theta\tan^2\theta\\$
Using $\tan\theta=\frac{\sin\theta}{\cos\theta}$, the above expression simplifies to
$=\cos^2\theta+\cos^2\theta\left(\frac{\sin^2\theta}{\cos^2\theta}\right) \\
=\cos^2\theta+\sin^2\theta\\
=1$
Since LHS=RHS, the identity's proof is complete.
