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Prove that $(\text{cosec} \theta + \sin \theta) (\text{cosec} \theta - \sin \theta) = \cot^2 \theta + \cos^2 \theta$.
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(a) LHS = $(\text{cosec} \theta + \sin \theta) (\text{cosec} \theta - \sin \theta) = \frac{(1+\sin^2 \theta)(1-\sin^2 \theta)}{\sin^2 \theta} = (1 + \sin^2 \theta) (\frac{\cos^2 \theta}{\sin^2 \theta}) = (\cot^2 \theta + \cos^2 \theta)$