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