Let S be the set of all solutions of the equation cos - 1 ( 2 x ) - 2 cos - 1 ( 1 - x 2 ) = , x[ - 1 2 , ? 1 2 ] . Then x S 2 sin - 1 ( x 2 - 1 ) is equal to [2023]

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Solution

Q.
Let $S$ be the set of all solutions of the equation $\cos^{- 1} (2 x) - 2 \cos^{- 1} (\sqrt{1 - x^{2}}) = π,$ $x \in [- \frac{1}{2}, \frac{1}{2}] .$ Then $\sum_{x \in S} 2 \sin^{- 1} (x^{2} - 1)$ is equal to [2023]

1 $0$

2 $\frac{- 2 π}{3}$

3 $π - 2 \sin^{- 1} (\frac{\sqrt{3}}{4})$

4 $π - \sin^{- 1} (\frac{\sqrt{3}}{4})$

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