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, $\tan \left[\frac{1}{2}{\cos }^{-1}\frac{\sqrt{5}}{3}\right]=\tan \frac{\theta }{2}=\frac{1-\cos \theta }{\sin \theta }=\frac{3-\sqrt{5}}{2}$