2 It is given that \(z = \mathrm { e } ^ { \mathrm { i } \theta }\), where \(0 < \theta < 2 \pi\), and \(w = \frac { 1 + z } { 1 - z }\).
- Prove that \(w = \mathrm { i } \cot \frac { 1 } { 2 } \theta\).
- Sketch separate Argand diagrams to show the locus of \(z\) and the locus of \(w\). You should show the direction in which each locus is described when \(\theta\) increases in the interval \(0 < \theta < 2 \pi\).