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Prove that : $\sqrt{\frac{\sec A-1}{\sec A+ 1}} + \sqrt{\frac{\sec A+ 1}{\sec A-1}} = 2 \cosec A$
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$LHS = \sqrt{\frac{\sec A-1}{\sec A+ 1}} + \sqrt{\frac{\sec A+ 1}{\sec A-1}}$
$= \frac{\sec A-1+ \sec A + 1}{\sqrt{\sec^2 A - 1}}$
$= \frac{2 \sec A}{\tan A}$
$= 2 \cosec A = RHS$
$= \frac{\sec A-1+ \sec A + 1}{\sqrt{\sec^2 A - 1}}$
$= \frac{2 \sec A}{\tan A}$
$= 2 \cosec A = RHS$