Question

Prove that $\sqrt{\frac{1-cosA}{1+cosA}}+\sqrt{\frac{1+cosA}{1-cosA}}=2cosecA$

Answer 1

LHS$=\sqrt{\frac{1-cosA}{1+cosA}}+\sqrt{\frac{1+cosA}{1-cosA}}$

Rationalize the denominators by multiplying suitable factors

$=\sqrt{\frac{1-\cos A}{1+\cos A}\times \frac{1-\cos A}{1-\cos A}}+\sqrt{\frac{1+\cos A}{1-\cos A}\times \frac{1+\cos A}{1+\cos A}}$

$=\frac{1-\cos A}{\sin A}+\frac{1+\cos A}{\sin A}$

$=\frac{2}{\sin A}$

$=2cosecA$

Hence proved