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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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LHS $= \frac{\sqrt{\sec A-1}}{\sqrt{\sec A+1}} + \frac{\sqrt{\sec A+1}}{\sqrt{\sec A-1}}$
$= \frac{(\sec A-1) + (\sec A+1)}{\sqrt{(\sec A+1)(\sec A-1)}}$
$= \frac{2 \sec A}{\sqrt{\sec^2 A-1}}$
$= \frac{2 \sec A}{\sqrt{\tan^2 A}}$
$= \frac{2 \sec A}{\tan A}$
$= \frac{2/\cos A}{\sin A/\cos A}$
$= \frac{2}{\sin A}$
$= 2 \operatorname{cosec} A = \text{RHS}$
$= \frac{(\sec A-1) + (\sec A+1)}{\sqrt{(\sec A+1)(\sec A-1)}}$
$= \frac{2 \sec A}{\sqrt{\sec^2 A-1}}$
$= \frac{2 \sec A}{\sqrt{\tan^2 A}}$
$= \frac{2 \sec A}{\tan A}$
$= \frac{2/\cos A}{\sin A/\cos A}$
$= \frac{2}{\sin A}$
$= 2 \operatorname{cosec} A = \text{RHS}$