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Prove the following trigonometric identity: $\frac{1 + \text{cosec } A}{\text{cosec } A} = \frac{\cos^2 A}{1 - \sin A}$
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$LHS = \frac{1 + \frac{1}{\sin A}}{\frac{1}{\sin A}} = \sin A + 1$
$= \frac{(\sin A + 1)(1 - \sin A)}{1 - \sin A}$
$= \frac{1 - \sin^2 A}{1 - \sin A}$
$= \frac{\cos^2 A}{1 - \sin A} = RHS$
$= \frac{(\sin A + 1)(1 - \sin A)}{1 - \sin A}$
$= \frac{1 - \sin^2 A}{1 - \sin A}$
$= \frac{\cos^2 A}{1 - \sin A} = RHS$