5 The complex numbers \(w\) and \(z\) are defined by \(w = 5 + 3 \mathrm { i }\) and \(z = 4 + \mathrm { i }\).
- Express \(\frac { \mathrm { i } w } { z }\) in the form \(x + \mathrm { i } y\), showing all your working and giving the exact values of \(x\) and \(y\).
- Find \(w z\) and hence, by considering arguments, show that
$$\tan ^ { - 1 } \left( \frac { 3 } { 5 } \right) + \tan ^ { - 1 } \left( \frac { 1 } { 4 } \right) = \frac { 1 } { 4 } \pi$$