Question

Value of the $\tan \left[\frac{1}{2}{\cos }^{-1}\left(\frac{2}{3}\right)\right]$ is

• A. $\sqrt{\frac{5}{2}}$
• B. $1-\sqrt{\frac{5}{2}}$
• C. $\frac{1}{\sqrt{5}}$
• D. $\sqrt{\frac{3}{10}}$

Answer 1

The correct option is C $\frac{1}{\sqrt{5}}$

Let
$\cos x=\frac{2}{3}\Rightarrow 2{\cos }^2\frac{x}{2}-1=\frac{2}{3}\Rightarrow \cos \frac{x}{2}=\frac{\sqrt{5}}{\sqrt{6}}$
So,
$\tan \left[\frac{1}{2}{\cos }^{-1}\left(\frac{2}{3}\right)\right]=\tan \left[\frac{x}{2}\right]=\frac{1}{\sqrt{5}}$