(i)[4]
On an Argand diagram with origin $O$, mark the points $A$, $B$ and $C$ for the complex numbers $u$, $u^*$ and $u^* - u$ respectively. What kind of quadrilateral is $OABC$?
(ii)[3]
Without a calculator and showing all working, write $\frac{u^*}{u}$ in the form $x + iy$, where $x$ and $y$ are real.
(iii)[3]
Using the argument of $\frac{u^*}{u}$, prove that $\tan^{-1}\!\left(\frac{3}{4}\right) = 2\tan^{-1}\!\left(\frac{1}{3}\right)$.