2021 EJC P2 Q2

It is given that \(z=\cos \theta +\mathrm{i}\sin \theta \), where \(0<\theta <\frac{\pi }{2}\).

1. Show that \({{\mathrm{e}}^{\mathrm{i}\left( \theta -\frac{\pi }{2} \right)}}=\sin \theta -\mathrm{i}\cos \theta \).[1]
2. Hence, or otherwise, show that \(\arg \left( 1-{{z}^{2}} \right)=\theta -\frac{\pi }{2}\) and find the modulus of \(1-{{z}^{2}}\).[3]
3. Hence, represent the complex number \(1-{{z}^{2}}\) on an Argand diagram.[2]
4. Given that \(\frac{{{z}^{*}}}{{{z}^{3}}\left( 1-{{z}^{2}} \right)}\) is real, find the possible values of \(\theta \).[3]