7 The complex number \(2 + \mathrm { i }\) is denoted by \(u\). Its complex conjugate is denoted by \(u ^ { * }\).
- Show, on a sketch of an Argand diagram with origin \(O\), the points \(A , B\) and \(C\) representing the complex numbers \(u , u ^ { * }\) and \(u + u ^ { * }\) respectively. Describe in geometrical terms the relationship between the four points \(O , A , B\) and \(C\).
- Express \(\frac { u } { u ^ { * } }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
- By considering the argument of \(\frac { u } { u ^ { * } }\), or otherwise, prove that
$$\tan ^ { - 1 } \left( \frac { 4 } { 3 } \right) = 2 \tan ^ { - 1 } \left( \frac { 1 } { 2 } \right) .$$