Given that ,
`tan^(-1) ((2x)/(1-x^(2))) = {{:(2 tan^(-1) x"," |x| le 1),(-pi +2 tan^(-1)x","x gt 1),(pi+2 tan^(-1)x"," x lt -1):}`
`sin^(-1)((2x)/(1+x^(2))) ={{:(2 tan^(-1)x","|x|le1),(pi -2 tan^(-1)x","x gt 1 and ),(-(pi+2tan^(-1))","x lt -1):}`
`sin^(-1) x + cos^(-1) x = pi//2 " for " - 1 le x le 1`
`sin^(-1) ((4x)/(x^(2)+4)) + 2 tan^(-1)( - x/2)` is independent of `x`, then

Explanation

Let `tan^(-1)x=theta(x=tantheta)impliestheta epsilon(-(pi)/2,(pi)/2)implies2thetaepsilon(-pi,pi)`
Now `sin^(-1)(2x)/(1+x^(2))=sin^(-1)sin 2theta={(-pi-2theta, 2thetaepsilon(-pi,-(pi)/2],"or",theta epsilon(-(pi)/2,-(pi)/2]),(2theta,2theta epsilon[-(pi)/2,(pi)/2],"or",theta epsilon[-(pi)/4,(pi)/4]),(pi-2theta,2theta epsilon[(pi)/2,pi),"or",theta epsilon[(pi)/4,(pi)/2)):}`
`=[(2tan^(-1)x,"if",x epsilon[-1,1]),(pi-2tan^(-1)x,"if",xge1),(-(pi+2tan^(-1)x),"if",xle-1):}`