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Prove that : $\frac{\sin \theta}{1 + \cos \theta} + \frac{1 + \cos \theta}{\sin \theta} = 2 \cosec \theta$
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$LHS = \frac{\sin^2 \theta + (1 + \cos \theta)^2}{(1 + \cos \theta) \sin \theta}$
$= \frac{\sin^2 \theta + 1 + 2 \cos \theta + \cos^2 \theta}{(1 + \cos \theta) \sin \theta}$
$= \frac{1 + 1 + 2 \cos \theta}{(1 + \cos \theta) \sin \theta}$
$= \frac{2 + 2 \cos \theta}{(1 + \cos \theta) \sin \theta}$
$= \frac{2 (1 + \cos \theta)}{(1 + \cos \theta) \sin \theta}$
$= \frac{2}{\sin \theta} = 2 \cosec \theta = RHS$
$= \frac{\sin^2 \theta + 1 + 2 \cos \theta + \cos^2 \theta}{(1 + \cos \theta) \sin \theta}$
$= \frac{1 + 1 + 2 \cos \theta}{(1 + \cos \theta) \sin \theta}$
$= \frac{2 + 2 \cos \theta}{(1 + \cos \theta) \sin \theta}$
$= \frac{2 (1 + \cos \theta)}{(1 + \cos \theta) \sin \theta}$
$= \frac{2}{\sin \theta} = 2 \cosec \theta = RHS$