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Number of solutions of tan^-1(ax) + tan^-1(bx) = a given angle

The same question, asked 2 times: 27 Jan 2024*, 22 Jan 2026 S1. Every wording below, oldest first.

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  1. Question 1

    27 Jan 2024

    151
    Considering only the principal values of inverse trigonometric functions, the number of positive real values of xx satisfying tan⁡−1(x)+tan⁡−1(2x)=π4\tan^{- 1}(x) +\tan^{- 1}(2x) =\frac{\pi}{4} is:

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  2. Question 2

    22 Jan 2026 S1

    68
    The number of solutions of tan⁡−14x+tan⁡−16x=π6\tan^{- 1}4x +\tan^{- 1}6x =\frac{\pi}{6}, where −126<x<126-\frac{1}{2\sqrt{6}}< x <\frac{1}{2\sqrt{6}} is equal to

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