Question

Solve : ${\cos }^{-1}\sqrt{\frac{1+\cos x}{2}}$

Answer 1

${\cos }^{-1}\sqrt{\frac{1+\cos x}{2}}={\cos }^{-1}\sqrt{\frac{{\cos }^2\frac{x}{2}+{\sin }^2\frac{x}{2}+{\cos }^2\frac{x}{2}-{\sin }^2\frac{x}{2}}{2}}$

$={\cos }^{-1}\sqrt{\frac{2{\cos }^2\frac{x}{2}}{2}}$

$={\cos }^{-1}\sqrt{{\cos }^2\frac{x}{2}}$

$={\cos }^{-1}\left(\cos \frac{x}{2}\right)$

$=\frac{x}{2}$

This is the required solution.