Solve for `x`: `tan^(-1)(2x)/(1-x^(2))+cot^(-1)(1-x^(2))/(2x))=pi/3, -1 lt x lt 1`.

Explanation

The given equation mya be written as
`tan^(-1)(2x)/(1-x^(2))+tan^(-1)(2x)/(1-x^(2))=pi/3`
`rArr 2tan^(-1)(2x)/(1-x^(2))=pi/3`
`rArr tan^(-1)(2x)/(1-x^(2))=pi/3`
`rArr (2x)/(1-x^(2))=tanpi/6=1/sqrt(3)`
`rArr (1-x^(2))=2sqrt(3)x`
`rArr x^(2)+2sqrt(3)x-1=0`
`rArr x=(-2sqrt(3)+-sqrt(12+4))/(2) = (-2sqrt(3)+-4)/(2)`
`=(-sqrt(3)+2)` or `(-sqrt(3)-2)`.
But, `-1 lt x lt 1`. So, `x ne (-sqrt(3),-2)`.
Hence, `x=(2-sqrt(3))`.