Prove that:
$$\frac{ an A}{1 - \cot A} + \frac{\cot A}{1 - an A} = 1 + \sec A \csc A$$

Question

Prove that:
$$\frac{	an A}{1 - \cot A} + \frac{\cot A}{1 - 	an A} = 1 + \sec A \csc A$$

Accepted Answer

Step 1: LHS $= \frac{\frac{\sin A}{\cos A}}{\frac{\sin A - \cos A}{\sin A}} + \frac{\frac{\cos A}{\sin A}}{\frac{\cos A - \sin A}{\cos A}}$
Step 2: $= \frac{1}{\sin A - \cos A}\left[\frac{\sin^2 A}{\cos A} - \frac{\cos^2 A}{\sin A}ight]$
Step 3: $= \frac{1}{\sin A - \cos A} 	imes \frac{(\sin A - \cos A)(\sin^2 A + \cos^2 A + \sin A \cos A)}{\sin A \cos A}$
Step 4: $= \frac{1 + \sin A \cos A}{\sin A \cos A} = \frac{1}{\sin A \cos A} + 1 = 1 + \sec A \csc A$ = RHS

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