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Prove that : $\sqrt{\frac{\sec A-1}{\sec A + 1}} + \sqrt{\frac{\sec A + 1}{\sec A - 1}} = 2 \operatorname{cosec} A$
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Sol. LHS $= \frac{\sec A - 1 + \sec A + 1}{\sqrt{\sec^2 A - 1}}$
$= \frac{2\sec A}{\tan A}$
$= 2\frac{1}{\cos A} \times \frac{\cos A}{\sin A}$
$= 2\operatorname{cosec} A = \operatorname{RHS}$
$= \frac{2\sec A}{\tan A}$
$= 2\frac{1}{\cos A} \times \frac{\cos A}{\sin A}$
$= 2\operatorname{cosec} A = \operatorname{RHS}$