Question

Evaluate $\tan \left[\frac{1}{2}{\cos }^{-1}\left(\frac{3}{\sqrt{11}}\right)\right]$

Answer 1

let $co{s}^{-1}\left(\frac{3}{\sqrt{1}1}\right)=x$

then $cosx=\frac{3}{\sqrt{1}1}$

now $sinx=\sqrt{1-co{s}^{2}x}=\sqrt{1-(\frac{3}{\sqrt{1}1}{)}^2}=\sqrt{\frac{2}{11}}$

now $tan\left[\frac{1}{2}co{s}^{-1}\right(\frac{3}{\sqrt{1}1}\left)\right]=\frac{1-cosx}{sinx}$

$=\frac{1-\frac{3}{\sqrt{1}1}}{\sqrt{\frac{2}{11}}}=\frac{\sqrt{11}-3}{\sqrt{2}}$