Prove that: `tan^(-1)((1-x)/(1+x))-tan^(-1)((1-y)/(1+y))=sin^(-1)((y-x)/(sqrt((1+x^2) (1+y^2))))`

Explanation

`tan^-1( (1-x)/(1+x)) - tan^-1 ((1-y)/(1+y))`

`tan^-1 (1) - tan^-1 x - [tan^-1 (1) - tan^-1 y]`

`= tan^-1 y - tan^-1 x`

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

`= sqrt(y^2 +x^2 -2xy+1+ x^2y^2 + 2xy)`

`= sqrt(y^2(1+x^2)+1(1+x^2))`

`= sqrt((y^2+1)(1+x^2)`

`sin^-1 ((y-x)/(sqrt(1+x^2)sqrt(1+y^2)))`

...