Prove that $\displaystyle \frac{\sin \mathrm{A}+\cos \mathrm{A}}{\sin \mathrm{A}-\cos \mathrm{A}}+\frac{\sin \mathrm{A}-\cos \mathrm{A}}{\sin \mathrm{A}+\cos \mathrm{A}}=\frac{2}{2 \sin ^{2} \mathrm{~A}-1}$

\[\begin{aligned} \text { L.H.S } & =\frac{(\sin A+\cos A)^{2}+(\sin A-\cos A)^{2}}{(\sin A-\cos A)(\sin A+\cos A)} \\ & =\frac{\sin ^{2} A+\cos ^{2} A+2 \sin A \cos A+\sin ^{2} A+\cos ^{2} A-2 \sin A \cos A}{\sin ^{2} A-\cos ^{2} A} \\ & =\frac{1+1}{\sin ^{2} A-\left(1-\sin ^{2} A\right)} \\ & =\frac{2}{2 \sin ^{2} A-1}=\text { R.H.S. } \end{aligned} \]