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← JEE Main 2022 · 28 Jul · Shift 1
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JEE Main 2022 · 28 Jul · Shift 1, Mathematics question 4

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 :

Choose one option
Answer and worked solution

Official answer: A

Rewrite. cos⁡−1x−2sin⁡−1x=π2−3θ\cos^{-1}x-2\sin^{-1}x=\frac\pi2-3\theta with θ=sin⁡−1x\theta=\sin^{-1}x, so sin⁡3θ=2sin⁡θ\sin3\theta=2\sin\theta.

Solve. sin⁡θ=0\sin\theta=0 or sin⁡2θ=14\sin^2\theta=\frac14: x=0,±12x=0,\pm\frac12, all of which check.

Sum 0. (A)

Topics:Inverse trigonometric functionsTrigonometric equations
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