<s>[INST] Prove equality of two numbers written in complex polar form. Show that these two numbers are equal: 
$$
z_1=\frac{e^{\tfrac{2\pi i}{9}}-e^{\tfrac{5\pi i}{9}}}{1-e^{\tfrac{7\pi i}{9}}}
$$
and
$$z_2=\frac{e^{\tfrac{\pi i}{9}}-e^{\tfrac{3\pi i}{9}}}{1-e^{\tfrac{4\pi i}{9}}}=\frac{1}{e^{\tfrac{\pi i}{9}}+e^{\tfrac{-\pi i}{9}}}.
$$
In case it helps, I do know that they are both real.
Thanks in advance for any suggestions!
 [/INST] Let $w = \exp(i\pi/9)$. Then
$$
z_1 = \frac{w^2-w^5}{1-w^7} = \frac{w^2(1-w^3)}{1-w^7}
$$
and
$$
z_2 = \frac{w-w^3}{1-w^4} = \frac{w(1-w^2)}{(1-w^2)(1+w^2)} = \frac{w}{1+w^2}.
$$
Hence
$$
z_1 - z_2 = \frac{w^2(1-w^3)(1+w^2)-w(1-w^7)}{(1-w^7)(1+w^2)} =
\frac{w(1-w)(w^6-w^3+1)}{(1-w^7)(1+w^2)}.
$$ 
It remains to show that $w^6 - w^3 + 1 = 0$, but
$$
w^6 - w^3 + 1 = w^{-3}(w^9 - w^6 + w^3) = w^{-3}(1-w^6+w^3)
$$
which establishes that. There are probably shorter ways to see this.
</s>