Write down all the roots of the equation \(x ^ { 5 } - 1 = 0\).
Use de Moivre's theorem to show that \(\cos 4 \theta = 8 \cos ^ { 4 } \theta - 8 \cos ^ { 2 } \theta + 1\).
Use the results of parts (a) and (b) to express each real root of the equation
$$8 x ^ { 9 } - 8 x ^ { 7 } + x ^ { 5 } - 8 x ^ { 4 } + 8 x ^ { 2 } - 1 = 0$$
in the form \(\cos k \pi\), where \(k\) is a rational number.