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Prove that $\frac{\sin A + \cos A}{\sin A - \cos A} + \frac{\sin A - \cos A}{\sin A + \cos A} = \frac{2}{1 - 2\cos^2 A}$.
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$$\begin{aligned}& LHS = \frac{(\sin A + \cos A)^2 + (\sin A - \cos A)^2}{\sin^2 A - \cos^2 A} \\ & = \frac{2(\sin^2 A + \cos^2 A)}{1 - \cos^2 A - \cos^2 A} \\ & = \frac{2}{1 - 2\cos^2 A} \\ & = RHS\end{aligned}$$