Question

Solve the following equation for x:

`tan^-1((2x)/(1-x^2))+cot^-1((1-x^2)/(2x))=(2pi)/3,x>0`

Answer 1

We know

`tan^-1x+tan^-1y=tan^-1((x+y)/(1-xy))`

`thereforetan^-1((2x)/(1-x^2))+cot^-1((1-x^2)/(2x))=(2pi)/3`

`=>tan^-1((2x)/(1-x^2))+tan^-1((2x)/(1-x^2))=(2pi)/3`    `[becausecot^1x=tan^-1  1/x]`

`=>tan^-1((2x)/(1-x^2))=pi/3`

`=>2tan^-1x=pi/3`     `[because2tan^-1xtan^-1((2x)/(1-x^2))]`

`=>tan^-1x=pi/6`

`=>x=tan  pi/6`

`=>x=1/sqrt3`