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Prove that :
$1 + \frac{\cot^2 \alpha}{1 + \operatorname{cosec} \alpha} = \operatorname{cosec} \alpha$
$1 + \frac{\cot^2 \alpha}{1 + \operatorname{cosec} \alpha} = \operatorname{cosec} \alpha$
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Sol. LHS $= 1 + \frac{\cot^2 \alpha}{1 + \operatorname{cosec} \alpha}$ (1 Mark)
$= 1 + \frac{(\operatorname{cosec} \alpha + 1)(\operatorname{cosec} \alpha - 1)}{(1 + \operatorname{cosec} \alpha)}$ (1/2 Mark)
$= 1 + \operatorname{cosec} \alpha - 1$ (1/2 Mark)
$= \operatorname{cosec} \alpha = \text{RHS}$
$= 1 + \frac{(\operatorname{cosec} \alpha + 1)(\operatorname{cosec} \alpha - 1)}{(1 + \operatorname{cosec} \alpha)}$ (1/2 Mark)
$= 1 + \operatorname{cosec} \alpha - 1$ (1/2 Mark)
$= \operatorname{cosec} \alpha = \text{RHS}$