Question

Show that `sin^(-1)(2x)/(1+x^(2))=[(2tan^(-1)x,"if",|x|le1),(pi-2tan^(-1)x,"if",xgt1),(-(pi+2tan^(-1)x),"if",xlt-1):}`

Answer 1

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):}`