Question

 

If `tan^(-1)((x-2)/(x-4)) +tan^(-1)((x+2)/(x+4))=pi/4` ,find the value of x

 

Answer 1

 

Given that

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

Taking LHS, we get:

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

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

`=tan^(-1)([((x-2)(x+4)(x+2)(x-4))/(x^2-16-(x^2-4))])`

`=tan^(-1)[(x^2+2x-8+x^2-2x-8)/(12)]`

`=tan^(-1)[(2x^2-16)/(-12)]`

hence

`tan^(-1)[(2x^2-16)/(-12)]=pi/4`

`[(2x^2-16)/(-12)]=tan (pi/4)`

`=>(x^2-8)/(-6)=1`

`=>x^2-8=-6`

`=>x^2=2`

`=>x=+-2`