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