If a+ib = (root of 1+i) / (root of 1-i )then prove a^2 +b^2 =1

Answer 1

a+bi = √(1+i)/√(1-i)
= √(1+i)/√(1-i) * √(1-i)/√(1-i)
= √(1 - i^2) / (1-i)
= √(1 - i^2) / (1-i) * (1+i)/(1+i)
= √2(1+i)/2
= √2/2 + √2/2 i

so a = √2/2 and b=√2/2
and a^2 + b^2 = 2/4 + 2/4 = 1