Question

Solve for x : tan^-1 (x - 1) + tan^-1x + tan^-1 (x + 1) = tan^-1 3x

Answer 1

Given that tan^-1(x-1)+tan^-1x+tan^-1(x-1)=tan^-13x

⇒ tan^-1(x-1)+tan^-1(x+1)=tan^-13x-tan^-1x ...(1)

We know that, tan^-1 A + tan^-1 B = tan-1 `((A+B)/(1-AB))` and tan^-1 A - tan^-1 B = tan-1`((A-B)/(1+AB))`

Thus, tan-1(x-1)+tan-1(x+1)=tan-1 `((x-1+x+1)/(1-(x-1)(x+1)))`

`=tan^(-1)((2x)/(1-(x^2-1)))`

`=tan^(-1)((2x)/(2-x^2)) `

Similarly, `tan^(-1)3x-tan^(-1)x=tan^(-1)((3x-x)/(1+3x(x)))`

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

From equations (1), (2) and (3), we have,

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

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

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

`=>2-x^2=1+3x^2`

`=>4x^2=1`

`=>x^2=1/4`

`=>x=+-1/2`