p289 q20

question: For any complex number z, let $\displaystyle \bar z$ and |z| be its conjugate and modulus respectively. suppose z_1 and z_2 are complex numbers such that $\displaystyle |z_1|=|z_2|=|z_1+z_2|=1$

(a)By using the property $\displaystyle z \bar z = |z|^2$, find the value of $\displaystyle (z_1+z_2) (\frac 1 z_1 + \frac 1 z_2)$

(b)Using the result of (a) , show that $\displaystyle z_1 ^2+z_1 z_2 +z_2^2 = 0$. Hence deduce that $\displaystyle z_1^3=z_2^3$

my working:

$\displaystyle =\frac {(z_1+z_2)^2} {z_1 z_2}$

$\displaystyle =\frac {z_1^2 + z_2^2 + 2z_1 z_2 }{z_1 z_2}$

$\displaystyle = 2 +\frac {z_1}{ z_2} + \frac{ z_2 }{z_1}$

stuck here and don't know how to do. thanks!