Question

Solve the following for x :

`tan^(-1)((x-2)/(x-3))+tan^(-1)((x+2)/(x+3))=pi/4,|x|<1`

Answer 1

 

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

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

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

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

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

`=>tan^(-1)((x-2)/(x-3))=tan^(-1)1/(2x+5)`

`=>(x-2)/(x-3)=1/(2x+5)`

`=>(x-2)(2x+5)=x-3`

`=>2x^2-4x+5x-10=x-3`

`=>2x^2=7`

`=>x=+-sqrt(7/2)`