Mark the tick against the correct answer in the following:
$\sin ^{-1}\left(\frac{-1}{2}\right)+2 \cos ^{-1}\left(\frac{-\sqrt{3}}{2}\right)=?$
A. $\frac{\pi}{2}$
B. $\pi$
C. $\frac{3 \pi}{2}$
D. none of these
To Find: The value of $\sin ^{-1}\left(\frac{-1}{2}\right)+2 \cos ^{-1}\left(\frac{-\sqrt{3}}{2}\right)$
Let, $x=\sin ^{-1}\left(\frac{-1}{2}\right)+2 \cos ^{-1}\left(\frac{-\sqrt{3}}{2}\right)$
$\Rightarrow x=-\sin ^{-1}\left(\frac{1}{2}\right)+2\left[\pi-\cos ^{-1}\left(\frac{\sqrt{3}}{2}\right)\right]\left(\because \sin ^{-1}(-\theta)=-\sin ^{-1}(\theta)\right.$ and $\left.\cos ^{-1}(-\theta)=\pi-\cos ^{-1}(\theta)\right)$
$\Rightarrow x=-\left(\frac{\pi}{6}\right)+2\left[\pi-\frac{\pi}{6}\right]$
$\Rightarrow x=-\left(\frac{\pi}{6}\right)+2\left[\frac{5 \pi}{6}\right]$
$\Rightarrow x=-\frac{\pi}{6}+\frac{5 \pi}{3}$
$\Rightarrow x=\frac{3 \pi}{2}$
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