Prove that
(cotA-cosecA)sq = 1-cosA/1+cosA

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Solution

$RHS = \frac{1 - \cos A}{1 + \cos A} = \frac{1 - \cos A}{1 + \cos A} \times \frac{1 - \cos A}{1 - \cos A} = \frac{1 + \cos^{2} A - 2 \cos A}{1 - \cos^{2} A} = \frac{1 + \cos^{2} A - 2 \cos A}{\sin^{2} A} = \frac{1}{\sin^{2} A} + \frac{\cos^{2} A}{\sin^{2} A} - 2 \times \frac{\cos A}{\sin A} \times \frac{1}{\sin A} = {cosec}^{2} A + \cot^{2} A - 2 \cot A . cosec A = {(\cot A - cosec A)}^{2} = LHS$

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