Question

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

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

Answer 1

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

We have, $\tan \left[\frac{1}{2}{\cos }^{-1}\left(\frac{2}{3}\right)\right]$
Put ${\cos }^{-1}\frac{2}{3}=\theta$
$\Rightarrow \cos \theta =\frac{2}{3}$
$\therefore \tan \left\{\frac{1}{2}{\cos }^{-1}\left(\frac{2}{3}\right)\right\}=\tan \frac{\theta }{2}$
$=\sqrt{\frac{1-\cos \theta }{1+\cos \theta }}=\sqrt{\frac{1-2/3}{1+2/3}}$
$=\sqrt{\frac{1/3}{5/3}}=\frac{1}{\sqrt{5}}$.