Question

Find the value of $\sin \left[2{\tan }^{-1}\left(\frac{4}{3}\right)\right]$

Answer 1

$\sin \left[2{\tan }^{-1}\left(\frac{4}{3}\right)\right]=t$

Let, ${\tan }^{-1}\left(\frac{4}{3}\right)=x$

$\tan x=\frac{4}{3}$.

(Refer to Image)

$\sin x=\frac{4}{5},\cos x\frac{3}{5}$

$\Rightarrow t=\sin 2x$

$\sin 2\theta =2\sin \theta \cos \theta$

$\Rightarrow t=2\sin x\cos x$

$t=2\left(\frac{4}{5}\right)\left(\frac{3}{5}\right)=\frac{24}{25}$