Prove that (1 + tan² A)/(1 + cot² A) = tan² A, and find the value of both sides at A = 60°.

To prove an identity, start with one side and change it step by step into the other. Write ratios in $\sin A$ and $\cos A$, use $\sin^2 A + \cos^2 A = 1$, $1 + \tan^2 A = \sec^2 A$ and $1 + \cot^2 A = \text{cosec}^2 A$, and factorise. One angle where a claim fails shows it is not an identity.