The value of $\sin [2 \tan^{-1}(0.75)]$ is

$0.96$

The correct answer is Option (3) → $0.96$ ##

We have, $\sin [2 \tan^{-1}(0.75)] = \sin \left[ 2 \tan^{-1} \frac{3}{4} \right] \dots(i)$

Let $\tan^{-1} \frac{3}{4} = x ⇒\tan x = \frac{3}{4}$

$∴\sin \left[ 2 \tan^{-1} \frac{3}{4} \right] = \sin[2x]$

$= \frac{2 \tan x}{1 + \tan^2 x}$

$= \frac{2 \times \frac{3}{4}}{1 + \left( \frac{3}{4} \right)^2} = \frac{\frac{3}{2}}{\frac{25}{16}}$

$= \frac{24}{25} = 0.96$