Prove that: $\sqrt{\frac{\csc heta - 1}{\csc heta + 1}} + \sqrt{\frac{\csc heta + 1}{\csc heta - 1}} = 2\sec heta$

Question

Prove that: $\sqrt{\frac{\csc	heta - 1}{\csc	heta + 1}} + \sqrt{\frac{\csc	heta + 1}{\csc	heta - 1}} = 2\sec	heta$

Accepted Answer

Step 1: LHS $= \sqrt{\frac{\csc	heta - 1}{\csc	heta + 1}} + \sqrt{\frac{\csc	heta + 1}{\csc	heta - 1}}$
$= \frac{(\csc	heta - 1) + (\csc	heta + 1)}{\sqrt{(\csc	heta + 1)(\csc	heta - 1)}}$
Step 2: $= \frac{2\csc	heta}{\sqrt{\csc^2	heta - 1}} = \frac{2\csc	heta}{\cot	heta}$
Step 3: $= \frac{2 \cdot \frac{1}{\sin	heta}}{\frac{\cos	heta}{\sin	heta}} = \frac{2}{\cos	heta} = 2\sec	heta$ = RHS

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