N2021 P1 Q4

Do not use a calculator in answering this question

The complex number \(z\) is given by

\(z=\frac{{{\left( \cos \left( \frac{1}{16}\pi \right)+\mathrm{i}\sin \left( \frac{1}{16}\pi \right) \right)}^{2}}}{\cos \left( \frac{1}{8}\pi \right)-\mathrm{i}\sin \left( \frac{1}{8}\pi \right)}\).

1. Find \(\left| z \right|\) and \(\arg \left( z \right)\). Hence find the value of \({{z}^{2}}\).[3]
2. 1. Show that \(\left( \cos \theta +\mathrm{i}\sin \theta \right)\left( 1+\cos \theta -\mathrm{i}\sin \theta \right)=1+\cos \theta +\mathrm{i}\sin \theta \).[2]
   2. Hence, or otherwise, find the value of \({{\left( 1+z \right)}^{4}}+{{\left( 1+{{z}^{*}} \right)}^{4}}\).[2]