Question

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 ___

Answer 1

The correct answer is 2.
Case I:x>0
tan−11−x22x​+tan−11−x22x​=3π​
x=2−3​
Case II : x<0
tan−11−x22x​+tan−11−x22x​+π=3π​
x=3​−1​⇒α=2