(a)[1]
Express $z\omega$ in the form $a + bi$, with $a$ and $b$ real numbers written exactly in surd form.
(b)[4]
Express $z$ and $\omega$ in the form $re^{i\theta}$, where $r > 0$ and $-\pi < \theta \leq \pi$. State the exact values of $r$ and $\theta$ for each one.
(c)[2]
On an Argand diagram, let $A$ and $B$ be the points corresponding to $\omega$ and $z\omega$ respectively. Prove that $OAB$ is an isosceles right-angled triangle, with $O$ as the origin.
(d)[3]
Using your results from part (b), prove that $\tan\left(\frac{5\pi}{12}\right) = \frac{\sqrt{3} + 1}{\sqrt{3} - 1}$.