If ^ -1 1 3 + ^ -1 2 3 = ^ -1 x, then x is equal to - Vidyadip

MCQ

If $\sin ^{-1} \frac{1}{3}+\sin ^{-1} \frac{2}{3}=\sin ^{-1} x$, then $x$ is equal to

A
$0$
B
$\frac{\sqrt{5}-4 \sqrt{2}}{9}$
✓
$\frac{\sqrt{5}+4 \sqrt{2}}{9}$
D
$\frac{\pi}{2}$

✓

Answer

Correct option: C.

$\frac{\sqrt{5}+4 \sqrt{2}}{9}$

(C) $\sin ^{-1} \frac{1}{3}+\sin ^{-1} \frac{2}{3}$
$=\sin ^{-1}\left[\frac{1}{3} \sqrt{1-\frac{4}{9}}+\frac{2}{3} \sqrt{1-\frac{1}{9}}\right]$
$=\sin ^{-1}\left[\frac{\sqrt{5}+4 \sqrt{2}}{9}\right]$
$\therefore \quad x=\frac{\sqrt{5}+4 \sqrt{2}}{9}$

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