Question

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

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

Answer 1

The correct option is C $\frac{1}{2}(3-\sqrt{5})$

Let $\alpha =co{s}^{-1}\frac{\sqrt{5}}{3}$
$cos\alpha =\frac{\sqrt{5}}{3}\Rightarrow sin\alpha =2$
$\therefore tan\left(\frac{1}{2}co{s}^{-1}\right(\frac{\sqrt{5}}{2}\left)\right)=tan\frac{\alpha }{2}=\frac{sin\frac{\alpha }{2}cos\frac{\alpha }{2}}{co{s}^2\frac{\alpha }{2}}=\frac{\frac{sin\alpha }{2}}{\frac{1+cos\alpha }{2}}$
$=\frac{sin\alpha }{1+cos\alpha }$
$=\frac{2/3}{1+\frac{\sqrt{5}}{3}}=\frac{2}{3+\sqrt{5}}=\frac{1}{2}(3-\sqrt{5})=\frac{3}{2}$