Question:
Evaluate $\sin \left(\tan ^{-1} \frac{3}{4}\right)$.
Solution:
We know that
$\tan ^{-1} x=\sin ^{-1} \frac{x}{\sqrt{1+x^{2}}}$
$\therefore \sin \left(\tan ^{-1} \frac{3}{4}\right)=\sin \left\{\sin ^{-1}\left(\frac{\frac{3}{4}}{\sqrt{1+\frac{9}{16}}}\right)\right\}$
$=\sin \left\{\sin ^{-1}\left(\frac{\frac{3}{4}}{\frac{5}{4}}\right)\right\}$
$=\sin \left(\sin ^{-1} \frac{3}{5}\right)$
$=\frac{3}{5} \quad\left[\because \sin \left(\sin ^{-1} x\right)=x\right]$
$\therefore \sin \left(\tan ^{-1} \frac{3}{4}\right)=\frac{3}{5}$
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