(2 ^ -1 (3 5 ) )= - Vidyadip

MCQ

$\sin \left(2 \sin ^{-1}\left(\frac{3}{5}\right)\right)=$

A
$\frac{24}{27}$
✓
$\frac{24}{25}$
C
$\frac{25}{24}$
D
$\frac{27}{24}$

✓

Answer

Correct option: B.

$\frac{24}{25}$

(B) Let $\theta=\sin ^{-1}\left(\frac{3}{5}\right)$
$\therefore \quad \sin \left(2 \sin ^{-1}\left(\frac{3}{5}\right)\right)=\sin 2 \theta$
$\begin{array}{l}=2 \sin \theta \cos \theta \\ =2 \sin \left(\sin ^{-1}\left(\frac{3}{5}\right)\right) \cos \left(\sin ^{-1}\left(\frac{3}{5}\right)\right)\end{array}$
$\begin{array}{l}=2 \times \frac{3}{5} \sqrt{1-\left(\frac{3}{5}\right)^2} \ldots\left[\because \cos \left(\sin ^{-1} x\right)=\sqrt{1-x^2}\right] \\ =2 \times \frac{3}{5} \times \frac{4}{5}=\frac{24}{25}\end{array}$

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