$\tan \left[\frac{1}{2} \cos ^{-1} \frac{4}{5}\right]=$
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) - $\frac{1}{3}$
$\tan \left[\frac{1}{2} \cos ^{-1} \frac{4}{5}\right]$
let $\frac{1}{2} \cos ^{-1} \frac{4}{5}=θ⇒\cos ^{-1} \frac{4}{5}=2θ$
$\frac{4}{5}=\cos 2θ$
$⇒\frac{4}{5}=\frac{1-\tan^2θ}{1+\tan^2θ}⇒4+4\tan^2θ=5-5\tan^2θ$
$9\tan^2θ=1$
$\tan^2θ=\frac{1}{9}$
$\tan θ=\sqrt{\frac{1}{9}}$
$\tan θ=\frac{1}{3}$