Let $S$ be the set of all solutions of the equation $\cos ^{-1}(2 x)-2 \cos ^{-1}\left(\sqrt{1-x^{2}}\right)=\pi, x \in\left[-\frac{1}{2}, \frac{1}{2}\right]$. Then $\sum_\limits{x \in S} 2 \sin ^{-1}\left(x^{2}-1\right)$ is equal to :

Question

$\pi-2 \sin ^{-1}\left(\frac{\sqrt{3}}{4}\right)$

$\pi-\sin ^{-1}\left(\frac{\sqrt{3}}{4}\right)$

$\frac{-2 \pi}{3}$

None

Solution

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