$\tan \left[\frac{1}{2} \cos ^{-1} \frac{4}{5}\right]=$

$\frac{1}{3}$

The correct answer is Option (1) - $\frac{1}{3}$

$\tan \left[\frac{1}{2} \cos ^{-1} \frac{4}{5}\right]$

let $\frac{1}{2} \cos ^{-1} \frac{4}{5}=θ⇒\cos ^{-1} \frac{4}{5}=2θ$

$\frac{4}{5}=\cos 2θ$

$⇒\frac{4}{5}=\frac{1-\tan^2θ}{1+\tan^2θ}⇒4+4\tan^2θ=5-5\tan^2θ$

$9\tan^2θ=1$

$\tan^2θ=\frac{1}{9}$

$\tan θ=\sqrt{\frac{1}{9}}$

$\tan θ=\frac{1}{3}$