(a) Write down all the roots of the equation \(x^{5}-1=0\).
(b) Use de Moivre's theorem to show that \(\cos 4 \theta=8 \cos ^{4} \theta-8 \cos ^{2} \theta+1\).
(c) 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.