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Prove that $\frac{\sin A + \cos A}{\sin A - \cos A} + \frac{\sin A - \cos A}{\sin A + \cos A} = \frac{2}{2 \sin^2 A-1}$
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L.H.S$= \frac{(\sin A+\cos A)^2+(\sin A-\cos A)^2}{(\sin A-\cos A)(\sin A+\cos A)}$
$= \frac{\sin^2 A+\cos^2 A+2\sin A\cos A+\sin^2 A+\cos^2 A-2\sin A\cos A}{\sin^2 A-\cos^2 A}$
$= \frac{1+1}{\sin^2 A-(1-\sin^2 A)}$
$= \frac{2}{2\sin^2 A-1} = \text{R.H.S.}$
$= \frac{\sin^2 A+\cos^2 A+2\sin A\cos A+\sin^2 A+\cos^2 A-2\sin A\cos A}{\sin^2 A-\cos^2 A}$
$= \frac{1+1}{\sin^2 A-(1-\sin^2 A)}$
$= \frac{2}{2\sin^2 A-1} = \text{R.H.S.}$