Answer: (a) The fourth roots of unity are \(1,-1,i,-i\).
(b) \(\cos 4\theta=8\cos^4\theta-8\cos^2\theta+1\).
(c) The real roots are
\(\cos\left(\frac{\pi}{24}\right),\ \cos\left(\frac{5\pi}{24}\right),\ \cos\left(\frac{7\pi}{24}\right),\ \cos\left(\frac{11\pi}{24}\right),\ \cos\left(\frac{13\pi}{24}\right),\ \cos\left(\frac{17\pi}{24}\right),\ \cos\left(\frac{19\pi}{24}\right),\ \cos\left(\frac{23\pi}{24}\right)\).
Equivalently, \(\pm\cos\left(\frac{\pi}{24}\right),\ \pm\cos\left(\frac{5\pi}{24}\right),\ \pm\cos\left(\frac{7\pi}{24}\right),\ \pm\cos\left(\frac{11\pi}{24}\right)\).
(a) The fourth roots of unity are the solutions of \(z^4=1\), namely
\(1,\ i,\ -1,\ -i\).
(b) By de Moivre's theorem,
\((\cos\theta+i\sin\theta)^4=\cos 4\theta+i\sin 4\theta\).
So \(\cos 4\theta\) is the real part of \((\cos\theta+i\sin\theta)^4\).
Expanding,
\((\cos\theta+i\sin\theta)^4=\cos^4\theta+4i\cos^3\theta\sin\theta-6\cos^2\theta\sin^2\theta-4i\cos\theta\sin^3\theta+\sin^4\theta\).
Hence
\(\cos 4\theta=\cos^4\theta-6\cos^2\theta\sin^2\theta+\sin^4\theta\).
Using \(\sin^2\theta=1-\cos^2\theta\),
\(\cos 4\theta=\cos^4\theta-6\cos^2\theta(1-\cos^2\theta)+(1-\cos^2\theta)^2\).
So
\(\cos 4\theta=\cos^4\theta-6\cos^2\theta+6\cos^4\theta+1-2\cos^2\theta+\cos^4\theta\)
\(=8\cos^4\theta-8\cos^2\theta+1\).
(c) Start with
\(16(8x^4-8x^2+1)^4-9=0\),
so
\((8x^4-8x^2+1)^4=\frac{9}{16}\).
Let \(x=\cos\theta\). Then from part (b),
\(8x^4-8x^2+1=8\cos^4\theta-8\cos^2\theta+1=\cos 4\theta\).
Therefore
\((\cos 4\theta)^4=\frac{9}{16}\),
hence
\(\cos 4\theta=\pm\frac{\sqrt3}{2}\).
So
\(4\theta=\pm\frac{\pi}{6}+2k\pi\quad\text{or}\quad 4\theta=\pm\frac{5\pi}{6}+2k\pi\).
Dividing by 4,
\(\theta=\frac{k\pi}{2}\pm\frac{\pi}{24}\quad\text{or}\quad \theta=\frac{k\pi}{2}\pm\frac{5\pi}{24}\).
Taking distinct cosine values gives the eight real roots
\(x=\cos\left(\frac{\pi}{24}\right),\ \cos\left(\frac{5\pi}{24}\right),\ \cos\left(\frac{7\pi}{24}\right),\ \cos\left(\frac{11\pi}{24}\right),\ \cos\left(\frac{13\pi}{24}\right),\ \cos\left(\frac{17\pi}{24}\right),\ \cos\left(\frac{19\pi}{24}\right),\ \cos\left(\frac{23\pi}{24}\right)\).
Equivalently,
\(x=\pm\cos\left(\frac{\pi}{24}\right),\ \pm\cos\left(\frac{5\pi}{24}\right),\ \pm\cos\left(\frac{7\pi}{24}\right),\ \pm\cos\left(\frac{11\pi}{24}\right)\).
