Question

if `tan^(-1)  (x-1)/(x - 2) + tan^(-1)  (x + 1)/(x + 2) = pi/4` then find the value of x.

Answer 1

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

`=> tan^(-1) [((x-1)/(x-2) + (x +1)/(x +2))/(1 - ((x-1)/(x-2))((x + 1)/(x+2)) ]] = pi/4`     `[tan^(-1) x + tan^(-1) y = tan^(-1)   (x+y)/(1-xy)]`

`=> tan^(-1) [((x-1)(x+2)+(x+1)(x-2))/((x + 2)(x-2) - (x - 1)(x + 1)]] = pi/4`

`=> tan^(-1) [(x^2 + x - 2 + x^2 -  x- 2)/(x^2 - 4 - x^2 + 1)] = pi/4`

`=> tan^(-1) [(2x^2 - 4)/(-3)] = pi/4`

`=> tan[tan^(-1)  (4 - 2x^2)/3] = tan  pi/4`

`=> (4- 2x^2)/3  = 1`

`=> 4  - 2x^2 = 3`

`=> 2x^2 = 4 - 3 =1`

`=> x = +- 1/sqrt2`

Hence, the value of x is  `+- 1/sqrt2`