Question

Find the value of `tan^-1  (x/y)-tan^-1((x-y)/(x+y))`

Answer 1

We know

`tan^-1x-tan^-1y=tan^-1  (x-y)/(1+xy),xy>1`

Now,

`tan^-1  (x/y)-tan^-1((x-y)/(x+y))`

`=tan^-1{(x/y-(x-y)/(x+y))/(1+x/y((x-y)/(x+y)))}`

`=tan^-1{((x^2+xy-xy+y^2)/(y(x+y)))/((x^2+y^2+xy-xy)/(y(x+y)))}`

`=tan^-1  1`

`=tan^-1(tan  pi/4)`

`=pi/4`

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