just write $z_1$ and $z_2$ in complex exponential form and the proof writes itself
u can write z1=r1*e^(theta1) z2=r2*e^(theta2) therefore dividing both these u get (r1/r2)*e^(theta1-theta2) so argument=(theta1-theta2)
now on rhs arg(z1)=theta1 and arg(z2)=theta2 so (theta1-theta2)
l.h.s=r.h.s
hence proved