Question

Prove that following: $\tan ^{-1} \sqrt{x}=\frac{1}{2} \cos ^{-1}\left(\frac{1-x}{1+x}\right), x \in(0,1)$

Answer 1

SOLUTION : R.H.S. $=\frac{1}{2} \cos ^{-1}\left(\frac{1-x}{1+x}\right)$. Put $x=\tan ^2 \theta \Rightarrow \tan \theta=\sqrt{x}$

$\therefore \theta=\tan ^{-1} \sqrt{x} ; \quad \text { R.H.S. }=\frac{1}{2}(2 \theta)=\theta=\tan ^{-1} \sqrt{x}$