Question

Solve the following equation for x:

`3sin^-1  (2x)/(1+x^2)-4cos^-1  (1-x^2)/(1+x^2)+2tan^-1  (2x)/(1-x^2)=pi/3`

Answer 1

`3sin^-1  (2x)/(1+x^2)-4cos^-1  (1-x^2)/(1+x^2)+2tan^-1  (2x)/(1-x^2)=pi/3`

`=>6tan^-1x-8tan^-1x+4tan^-1x=pi/3`     `[because 2tan^-1x=sin^-1((2x)/(1+x^2)),2tan^-1x=cos^-1((1-x^2)/(1+x^2))and 2tan^-1x=tan^-1((2x)/(1-x^2))]`

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

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

`=>x=tan  pi/6`

`=>x=1/sqrt3`