Prove that:
$$\frac{ an heta}{1 - \cot heta} + \frac{\cot heta}{1 - an heta} = 1 + \sec heta \cosec heta$$

Question

Prove that:
$$\frac{	an 	heta}{1 - \cot 	heta} + \frac{\cot 	heta}{1 - 	an 	heta} = 1 + \sec 	heta \cosec 	heta$$

Accepted Answer

Step 1: LHS $= \frac{\frac{\sin 	heta}{\cos 	heta}}{1 - \frac{\cos 	heta}{\sin 	heta}} + \frac{\frac{\cos 	heta}{\sin 	heta}}{1 - \frac{\sin 	heta}{\cos 	heta}}$
Step 2: $= \frac{\sin^2 	heta}{\cos 	heta(\sin 	heta - \cos 	heta)} + \frac{\cos^2 	heta}{\sin 	heta(\cos 	heta - \sin 	heta)}$
Step 3: $= \frac{\sin^3 	heta - \cos^3 	heta}{\sin 	heta \cos 	heta(\sin 	heta - \cos 	heta)}$
Step 4: $= \frac{\sin^2 	heta + \sin 	heta \cos 	heta + \cos^2 	heta}{\sin 	heta \cos 	heta} = \frac{1}{\sin 	heta \cos 	heta} + 1 = 1 + \sec 	heta \cosec 	heta$ = RHS

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