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  4. If the sum of all the solutions of tan -1((2 x/1-x2))+ cot -1((1-x2/2 x))=(π/3),-1 < x < 1, x ≠ 0, is α-(4/√3), then α is equal to

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Q. If the sum of all the solutions of $\tan ^{-1}\left(\frac{2 x}{1-x^2}\right)+\cot ^{-1}\left(\frac{1-x^2}{2 x}\right)=\frac{\pi}{3},-1 < x < 1, x \neq 0$, is $\alpha-\frac{4}{\sqrt{3}}$, then $\alpha$ is equal to ____

JEE MainJEE Main 2023Inverse Trigonometric Functions

Solution:

Case $I : x >0$
$ \tan ^{-1} \frac{2 x}{1-x^2}+\tan ^{-1} \frac{2 x}{1-x^2}=\frac{\pi}{3}$
$x=2-\sqrt{3}$
Case II : $x < 0$
$ \tan ^{-1} \frac{2 x}{1-x^2}+\tan ^{-1} \frac{2 x}{1-x^2}+\pi=\frac{\pi}{3} $
$x=\frac{-1}{\sqrt{3}} \Rightarrow \alpha=2$