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sin^-1(2x sqrt(1 - x^2)) in terms of cos^-1 x on a given range

This type of question, asked 2 times: 15 May 2023 S2, 14 May 2026 S2. All 2 questions below, oldest first.

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  1. Question 1

    15 May 2023 S2

    124
    If cos⁡−1x=α (0<x<1)\cos^{-1}x = \alpha\,(0 < x < 1) and sin⁡−1(2x1−x2)+sec⁡−1(12x2−1)=2π3\sin^{-1}(2x\sqrt{1-x^2}) + \sec^{-1}\left(\frac{1}{2x^2-1}\right) = \frac{2\pi}{3}, then α\alpha is

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  2. Question 2

    14 May 2026 S2

    68
    If −1≤x≤−12-1 \le x \le -\dfrac{1}{\sqrt{2}} then sin⁡−1(2x1−x2)=\sin^{-1}\left(2x\sqrt{1 - x^2}\right) =

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