Question

Find the value of: `tan1/2[sin^(-1)(2x)/(1+x^2)+cos^(-1)(1-y^2)/(1+y^2)],|x|<1, y >0` and `x y" "<" "1`

Answer 1

`tan""(1)/(2)[sin^(-1)""(2x)/(1+x^(2))+cos^(-1)""(1-y^(2))/(1+y^(2))]`
`=tan{(1)/(2)(2 tan^(-1)x+2 tan^(-1)y)}" " { :. 2 tan^(-1)=sin^(-1)""(2x)/(1+x^(2)) and 2 tan^(-1)y=cos^(-1)""(1-y^(2))/(1-xy)}`
`=tan(tan^(-1)x+tan^(-1)y)`
`=tan(tan^(-1)""(x+y)/(1-xy))=(x+y)/(1-xy)`