Question

Evaluate: $\tan \left[\frac{1}{2}{\cos }^{-1}\left(\frac{\sqrt{5}}{3}\right)\right]$

• A. $\frac{3+\sqrt{5}}{2}$
• B. $\frac{3-\sqrt{5}}{2}$
• C. $\frac{4+\sqrt{5}}{2}$
• D. $\frac{4-\sqrt{5}}{2}$

Answer 1

The correct option is B $\frac{3-\sqrt{5}}{2}$

Let, $\frac{1}{2}{\cos }^{-1}\left(\frac{\sqrt{5}}{3}\right)=\theta$.......(1)

$\Rightarrow \cos 2\theta =\frac{\sqrt{5}}{3}$

$\Rightarrow \frac{1-{\tan }^2\theta }{1+{\tan }^2\theta }$$=\frac{\sqrt{5}}{3}$

$\Rightarrow \frac{1}{{\tan }^2\theta }$$=\frac{3+\sqrt{5}}{3-\sqrt{5}}$

$\Rightarrow {\tan }^2\theta =\frac{3-\sqrt{5}}{3+\sqrt{5}}=\frac{(3-\sqrt{5}{)}^2}{4}$

$\therefore \tan \theta =\frac{3-\sqrt{5}}{2}$.

According to the problem and from (1) it follows that we are to find the value of $\tan \theta$ and which is equal to $\frac{3-\sqrt{5}}{2}$.