Question

The value of $\tan \left(\frac{1}{2} \cos ^{-1} \frac{\sqrt{5}}{3}\right)$ is

• A. $\frac{3+\sqrt{5}}{2}$
• B. $\frac{3-\sqrt{5}}{2}$
• C. $\frac{\sqrt{5}}{6}$
• D. None of these

Answer 1

Answer: (B)

Let $\cos ^{-1} \frac{\sqrt{5}}{3}=\theta$, then $0<\theta<\frac{\pi}{2}$ and $\cos \theta=\frac{\sqrt{5}}{3}$
$\therefore \sin \theta=\sqrt{1-\cos ^{2} \theta}=\frac{2}{3}$
So, $\tan \left[\frac{1}{2} \cos ^{-1} \frac{\sqrt{5}}{3}\right]=\tan \frac{\theta}{2}=\frac{1-\cos \theta}{\sin \theta}=\frac{3-\sqrt{5}}{2}$