$\dfrac{1}{1 + \tan^2 A} + \dfrac{1}{1 + \cot^2 A} = ?$

Using $1 + \tan^2 A = \sec^2 A$ and $1 + \cot^2 A = \operatorname{cosec}^2 A$, the sum becomes $\cos^2 A + \sin^2 A = 1$.