Question

Integrate $\int {\tan }^{-1}\left(\sqrt{\frac{1-\cos 2x}{1+\cos 2x}}\right)dx$

Answer 1

Applying the trigonometric identities in $\int {\tan }^{-1}\left(\sqrt{\frac{1-\cos 2x}{1+\cos 2x}}\right)dx$.

$\int {\tan }^{-1}\left(\sqrt{\frac{1-\cos 2x}{1+\cos 2x}}\right)dx=\int {\tan }^{-1}\left(\sqrt{\frac{2{\sin }^{2}x}{2{\cos }^{2}x}}\right)dx$

$=\int {\tan }^{-1}\left(\tan x\right)dx$

$=\int xdx$

$=\frac{x^2}{2}+c$