8 The complex number \(\omega\) is given by \(\omega = \cos \frac { 2 \pi } { 5 } + \mathrm { i } \sin \frac { 2 \pi } { 5 }\).
- Verify that \(\omega\) is a root of the equation \(z ^ { 5 } = 1\).
- Write down the three other non-real roots of \(z ^ { 5 } = 1\), in terms of \(\omega\).
- Show that \(1 + \omega + \omega ^ { 2 } + \omega ^ { 3 } + \omega ^ { 4 } = 0\).
- Hence show that \(\left( \omega + \frac { 1 } { \omega } \right) ^ { 2 } + \left( \omega + \frac { 1 } { \omega } \right) - 1 = 0\).
- Hence show that \(\cos \frac { 2 \pi } { 5 } = \frac { \sqrt { 5 } - 1 } { 4 }\).