8 The variable complex number \(z\) is given by
$$z = 1 + \cos 2 \theta + i \sin 2 \theta$$
where \(\theta\) takes all values in the interval \(- \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi\).
- Show that the modulus of \(z\) is \(2 \cos \theta\) and the argument of \(z\) is \(\theta\).
- Prove that the real part of \(\frac { 1 } { z }\) is constant.