6 It is given that \(z = \mathrm { e } ^ { \mathrm { i } \theta }\).
- Show that
$$z + \frac { 1 } { z } = 2 \cos \theta$$
- Find a similar expression for
$$z ^ { 2 } + \frac { 1 } { z ^ { 2 } }$$
(2 marks)
- Hence show that
$$z ^ { 2 } - z + 2 - \frac { 1 } { z } + \frac { 1 } { z ^ { 2 } } = 4 \cos ^ { 2 } \theta - 2 \cos \theta$$
(3 marks)
- Hence solve the quartic equation
$$z ^ { 4 } - z ^ { 3 } + 2 z ^ { 2 } - z + 1 = 0$$
giving the roots in the form \(a + \mathrm { i } b\).