Inverse Trigonometric Functions

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 :