Answer :
To prove the identity we need to remember the following identities:
[tex]\cos ^2\theta+\sin ^2\theta=1[/tex][tex]\tan \theta=\frac{\sin \theta}{\cos \theta}[/tex]With this in mind we have:
[tex]\begin{gathered} \frac{\cos^2\theta+\sin^2\theta}{1+\tan^2\theta}=\frac{1}{1+\frac{\sin ^2\theta}{\cos ^2\theta}} \\ =\frac{1}{\frac{\cos ^2\theta+\sin ^2\theta}{\cos ^2\theta}} \\ =\frac{1}{\frac{1}{\cos ^2\theta}} \\ =\cos ^2\theta \end{gathered}[/tex]This proves the identity.