(i) $\sin ^{-1} \frac{1}{2}-2 \sin ^{-1} \frac{1}{\sqrt{2}}$
(ii) $\sin ^{-1}\left\{\cos \left(\sin ^{-1} \frac{\sqrt{3}}{2}\right)\right\}$
(i)
$\sin ^{-1} \frac{1}{2}-2 \sin ^{-1} \frac{1}{\sqrt{2}}=\sin ^{-1} \frac{1}{2}-\sin ^{-1} 2 \times \frac{1}{\sqrt{2}} \sqrt{1-\left(\frac{1}{\sqrt{2}}\right)^{2}}$
$=\sin ^{-1} \frac{1}{2}-\sin ^{-1} \sqrt{2} \times \frac{1}{\sqrt{2}}$
$=\sin ^{-1} \frac{1}{2}-\sin ^{-1} 1$
$=\sin ^{-1}\left(\sin \frac{\pi}{6}\right)-\sin ^{-1}\left(\sin \frac{\pi}{2}\right)$
$=\frac{\pi}{6}-\frac{\pi}{2}$
$=-\frac{\pi}{3}$
(ii)
$\sin ^{-1}\left\{\cos \left(\sin ^{-1} \frac{\sqrt{3}}{2}\right)\right\}=\sin ^{-1}\left\{\cos \left(\sin ^{-1} \sin \frac{\pi}{3}\right)\right\}$
$=\sin ^{-1}\left\{\cos \left(\frac{\pi}{3}\right)\right\}$
$=\sin ^{-1}\left\{\frac{1}{2}\right\}$
$=\sin ^{-1}\left\{\sin \frac{\pi}{6}\right\}$
$=\frac{\pi}{6}$
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