Trouble figuring out a complex number equation I'm struggling to prove this question, an help is greatly appreciated!
If $a=cis(\pi/5)$, prove that:
$$a^7=-a^2$$
and
$$a^9=-a^4$$

Let's think about it geometrically. $\frac{7 \pi}{5}$ and $\frac{2\pi}{5}$ differ by $\pi$, and so do $\frac{9 \pi}{5}$ and $\frac{4\pi}{5}$.
Geometrically, this means that in the complex plane, $e^{ i\frac{7 \pi}{5}}$ and $e^{i \frac{2\pi}{5}}$ point in opposite directions. Written algebraically, we have $e^{ i\frac{7 \pi}{5}} = - e^{i \frac{2\pi}{5}}$, or $a^7 = - a^2$.
Similarly, $e^{i \frac{9 \pi}{5}}$ and $e^{i \frac{4\pi}{5}}$ point in opposite directions, so $e^{i \frac{9 \pi}{5}} = - e^{i \frac{4\pi}{5}}$, or $a^9 = - a^4$.
