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Solve an equation in sin^-1 x and cos^-1 x using sin^-1 x + cos^-1 x = pi/2

This type of question, asked 4 times: 28 Jul 2022, 9 Apr 2024, 29 Jan 2025, 28 Jan 2026 S1. All 4 questions below, oldest first.

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  1. Question 1

    28 Jul 2022

    64
    Considering the principal values of the inverse trigonometric functions, the sum of all the solutions of the equation cos⁡−1(x)−2sin⁡−1(x)=cos⁡−1(2x)\cos^{- 1}(x) - 2\sin^{- 1}(x) =\cos^{- 1}(2x) is equal to:

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  2. Question 2

    9 Apr 2024

    179
    Let the inverse trigonometric functions take principal values. The number of real solutions of the equation 2sin⁡−1x+3cos⁡−1x=2π52\sin^{- 1}x+ 3\cos^{- 1}x=\frac{2\pi}{5}, is

  3. Question 3

    29 Jan 2025

    75
    Let S={x:cos⁡−1x=π+sin⁡−1x+sin⁡−1(2x+1)}S=\{x:\cos^{-1}x=\pi+\sin^{-1}x+\sin^{-1}(2x+1)\}. Then ∑x∈S(2x−1)2\sum_{x\in S}(2x-1)^{2} is equal to

  4. Question 4

    28 Jan 2026 S1

    73
    If k=tan⁡(π4+12cos⁡−1(23))+tan⁡(12sin⁡−1(23))k = \tan\left( \frac{\pi}{4} + \frac{1}{2}\cos^{- 1}\left( \frac{2}{3} \right) \right) + \tan\left( \frac{1}{2}\sin^{- 1}\left( \frac{2}{3} \right) \right) then the number of solutions of the equation sin⁡−1(kx−1)=sin⁡−1x−cos⁡−1x\sin^{- 1}(kx - 1) = \sin^{- 1}x - \cos^{- 1}x is......... .