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Equation with 2tan^-1 x written as tan^-1(2x/(1-x^2)) or cos^-1((1-x^2)/(1+x^2)), branch by branch

This type of question, asked 2 times: 25 Jan 2023, 1 Feb 2023. All 2 questions below, oldest first.

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  1. Question 1

    25 Jan 2023

    85
    If the sum of all the solutions of tan⁡−1(2x1−x2)+cot⁡−1(1−x22x)=π3,−1<x<1,x≠0\tan^{- 1}\left( \frac{2x}{1 -x^{2}} \right)+\cot^{- 1}\left( \frac{1 -x^{2}}{2x} \right)=\frac{\pi}{3}, - 1 < x < 1,x \neq 0, is α−43\alpha -\frac{4}{\sqrt{3}}, then α\alpha is equal to

  2. Question 2

    1 Feb 2023

    166
    Let S={x∈R:0<x<1S =\left\{ x \in \mathbb{R}:0 < x < 1 \right. and  2tan⁡−1(1−x1+x)=cos⁡−1(1−x21+x2)}\left. \ 2\tan^{- 1}\left( \frac{1 - x}{1 + x} \right)=\cos^{- 1}\left( \frac{1 -x^{2}}{1 +x^{2}} \right) \right\}. If n(S)n(S) denotes the number of elements in SS then:

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