Question

Write the following functions in the simplest form : $\tan ^{-1} \sqrt{\frac{1-\cos x}{1+\cos x}}, 0<x<\pi$.

Answer 1

SOLUTION : Let $y=\tan ^{-1} \sqrt{\frac{1-\cos x}{1+\cos x}} \Rightarrow y=\tan ^{-1} \sqrt{\frac{2 \sin ^2 \frac{x}{2}}{2 \cos ^2 \frac{x}{2}}}=\tan ^{-1}|\tan x / 2| \quad\left[\because 0<\frac{x}{2}<\frac{\pi}{2}\right]$