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1/cosecA -Cot A - 1/sinA = 1/ sin A - 1/cosec A + cot A

$Given \frac{1}{\csc A - \cot A} - \frac{1}{\sin A} = \frac{1}{\csc A - \cot A} \times \frac{\csc A + \cot A}{\csc A + \cot A} - \frac{1}{\sin A} = \frac{\csc A + \cot A}{\csc^{2} A - \cot^{2} A} - \csc A = \csc A + \cot A - \csc A = \cot A \frac{1}{\sin A} - \frac{1}{\csc A + \cot A} \times \frac{\csc A - \cot A}{\csc A - \cot A} = \csc A - \frac{(\csc A - \cot A)}{\csc^{2} A - \cot^{2} A} = \csc A - \csc A + \cot A = \cot A \Rightarrow \frac{1}{\csc A - \cot A} - \frac{1}{\sin A} = \frac{1}{\sin A} - \frac{1}{\csc A + \cot A}$

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