If a2+b2=1,  then 1+b+ia1+b−ia  =

Given that a2+b2=1.  Therefore, 1+b+ia1+b−ia   =   (1+b+ia)(1+b+ia)(1+b−ia) (1+b+ia)                     =(1+b)2−a2+2ia(1+b)1+b2+2b+a2                     =(1−a2)+2b+b2+2ia(1+b)2(1+b)                     =2b2+2b+2ia(1+b)2(1+b)                     =b  +  ia