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Prove that : $\frac{\sec^3 \theta}{\sec^2 \theta - 1} + \frac{\cosec^3 \theta}{\cosec^2 \theta - 1} = \sec \theta \cdot \cosec \theta (\sec \theta + \cosec \theta)$
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LHS = $\frac{\sec^3 \theta}{(\sec^2 \theta - 1)} + \frac{\cosec^3 \theta}{(\cosec^2 \theta - 1)}$
$= \frac{\sec^3 \theta}{\tan^2 \theta} + \frac{\cosec^3 \theta}{\cot^2 \theta}$ (1 Mark)
$= \frac{1}{\cos^3 \theta} \times \frac{\cos^2 \theta}{\sin^2 \theta} + \frac{1}{\sin^3 \theta} \times \frac{\sin^2 \theta}{\cos^2 \theta}$ (1/2 Mark)
$= \frac{1}{\cos \theta \sin^2 \theta} + \frac{1}{\sin \theta \cos^2 \theta}$ (1/2 Mark)
$= \frac{1}{\sin \theta \cos \theta} [\frac{1}{\sin \theta} + \frac{1}{\cos \theta}]$ (1/2 Mark)
$= \sec \theta \cdot \cosec \theta (\sec \theta + \cosec \theta) = \text{RHS}$ (1/2 Mark)
$= \frac{\sec^3 \theta}{\tan^2 \theta} + \frac{\cosec^3 \theta}{\cot^2 \theta}$ (1 Mark)
$= \frac{1}{\cos^3 \theta} \times \frac{\cos^2 \theta}{\sin^2 \theta} + \frac{1}{\sin^3 \theta} \times \frac{\sin^2 \theta}{\cos^2 \theta}$ (1/2 Mark)
$= \frac{1}{\cos \theta \sin^2 \theta} + \frac{1}{\sin \theta \cos^2 \theta}$ (1/2 Mark)
$= \frac{1}{\sin \theta \cos \theta} [\frac{1}{\sin \theta} + \frac{1}{\cos \theta}]$ (1/2 Mark)
$= \sec \theta \cdot \cosec \theta (\sec \theta + \cosec \theta) = \text{RHS}$ (1/2 Mark)