Question

Solve the following equation for x:

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

Answer 1

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

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

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

⇒ `((x-2)(x+1)+(x-1)(x+2))/((x-1)(x+1)-(x-2)(x+2))=1`

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

⇒ `(2x^2-4)/3=1`

⇒ `2x^2-4=3`

⇒ `2x^2=7`

⇒ `x^2=7/2`

∴ `x=+-sqrt(7/2`