(a)OR(b)

Prove that: $\displaystyle \sqrt{\frac{\sec \mathrm{A}-1}{\sec \mathrm{~A}+1}}+\sqrt{\frac{\sec \mathrm{A}+1}{\sec \mathrm{~A}-1}}=2 \operatorname{cosec} \mathrm{~A}$

LHS \(\displaystyle =\frac{\sec \mathrm{A}-1+\sec \mathrm{A}+1}{\sqrt{\sec ^{2} \mathrm{~A}-1}}\) \(\displaystyle =\frac{2 \sec \mathrm{~A}}{\tan \mathrm{~A}}\) \(\displaystyle =2 \operatorname{cosec} \mathrm{~A}=\mathrm{RHS}\) LHS \(\displaystyle =\left(\frac{1-\cos ^{2} \mathrm{~A}}{\cos \mathrm{~A}}\right)\left(\frac{1-\sin ^{2} \mathrm{~A}}{\sin \mathrm{~A}}\right)\) \(\displaystyle =\frac{\sin ^{2} \mathrm{~A}}{\cos \mathrm{~A}} \cdot \frac{\cos ^{2} \mathrm{~A}}{\sin \mathrm{~A}}\) \(\displaystyle =\sin \mathrm{A} . \cos \mathrm{A}\) RHS \(\displaystyle =\frac{\text { sin } \mathrm{A} \cdot \cos \mathrm{A}}{\sin ^{2} \mathrm{~A}+\cos ^{2} \mathrm{~A}}\) \(\displaystyle =\sin \mathrm{A} \cdot \cos \mathrm{A}\) \(\displaystyle \therefore \mathrm{LHS}=\mathrm{RHS}\)