Z2^2=cos2a+sin2a

z 1 =√3+I = 2(cosπ/6+isπ/6)

Z1* (z2) 2 = 2 [cos (2a+π/6)+isin (2a+π/6)] is a pure imaginary number with negative imaginary part, so

2a+π /6=3π /2

a=2π /3

Z2=- 1/2+i√3/2