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