We have to prove that |(a-b)| = |(1-ab*)| , where |a| = 1

Hence, we need to show that the magnitude of the LHS = RHS

|a| = aa* = 1 => a = 1/a*

Hence,

LHS = |(a-b)|

RHS = |(1-b*/a*)| => |(a*-b*)|/|a*| = |(a*-b*)|/1 = |(a*-b*)|

============> LHS = |(a-b)| = |(a*-b*)| = RHS ==================> It is PROVEN.

Since a & b are distinct complex numbers,

a = b + ci ; a* = b - ci ; b = x +yi ; b* = x - yi

LHS = |(a-b)| = |(b+ci-x-yi )| = |(b-x)+(c-y)i| = [(b-x)^2 + (c-y)^2]^(1/2)

RHS = |(a*-b*)| = |(b-ci-x+yi)| = |(b-x)+(y-c)i| = [(b-x)^2 + (y-c)^2]^(1/2)

Note: (c -y) = [-(c-y)]

let (c - y) be h,

h = -h => h^2 = (-h)^2 = |h|^2

Hence,

let (b-x) be p,

LHS = (p^2 + h^2)^(1/2)

RHS = (p^2 + h^2)^(1/2) = LHS <=========== PROVEN