Argument of complex number z3? z3=(z1/z2)^2; z1=1-sqrt(3)*i; z2=sqrt(3)+i

(z₁)^2 = (1 - i√3)^2 = (1 - 2i√3 - 3) = (- 2 - 2i√3) = -2(1 + 2i√3)

(z₂)^2 = (√3 + i)^2 = (3 +2i√3 - 1) =   ( 2 + 2i√3) = 2( 1 + 2i√3)

So ......

[(z₁)^2] / [(z₂)^2] = [ -2(1 + 2i√3)] / [2(1 + 2i√3)]  = -2/2 = -1  =  -1 + 0i  = z₃

So

tan Θ = (b/a)  →  tan^-1 (b/a) = Θ  = arg z₃

And

tan^-1 (0/-1) = pi

So

arg z₃ = pi