write them down as exponential funcitons: z1=r1*exp(i*phi1)

calculate the |z1+z2| by multiplying by the same with -i instead of i

r1 and r2 all fall out

it stays exp(i*(phi1-phi2)) = - exp(-i*(phi1-phi2))

divide by exp(-i*(phi1-phi2))

now it's exp(i*2*(phi1-phi2)) = -1

so phi1-phi2 must be pi/2 because exp(i*2*pi/2)=exp(i*pi)=-1