6 The complex number \(u\) is defined by
$$u = \frac { 7 + \mathrm { i } } { 1 - \mathrm { i } }$$
- Express \(u\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
- Show on a sketch of an Argand diagram the points \(A , B\) and \(C\) representing \(u , 7 + \mathrm { i }\) and \(1 - \mathrm { i }\) respectively.
- By considering the arguments of \(7 + \mathrm { i }\) and \(1 - \mathrm { i }\), show that
$$\tan ^ { - 1 } \left( \frac { 4 } { 3 } \right) = \tan ^ { - 1 } \left( \frac { 1 } { 7 } \right) + \frac { 1 } { 4 } \pi$$