Question

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

Answer 1

Consider the following question.

$y=\tan \left(\frac{1}{2}{\cos }^{-1}\left(\frac{\sqrt{5}}{3}\right)\right)...\left(1\right)$

$x={\cos }^{-1}\left(\frac{\sqrt{5}}{3}\right)$

$\cos x=\left(\frac{\sqrt{5}}{3}\right)$

Using a Pythagoras theorem,

$\tan x=\frac{2}{3}$

$x={\tan }^{-1}\left(\frac{2}{3}\right)$

$y=\tan \left(\frac{1}{2}{\tan }^{-1}\left(\frac{2}{3}\right)\right)$

$=\frac{1}{3}$

Hence, this is the required answer.