Compute \(\cos^4\frac{\pi}{9}+\cos^4\frac{2\pi}{9}+\cos^4\frac{3\pi}{9}+\cos^4\frac{4\pi}{9}\).
Solution
Answer: \(\frac{19}{16}\)
Note that \(\frac\pi9,\frac{7\pi}9,\frac{13\pi}9\) satisfy \(\cos3\theta=\frac12\). Let \(r=\cos\frac\pi9\), \(s=\cos\frac{7\pi}9\), and \(t=\cos\frac{13\pi}9\). Then \(r,s,t\) are the three roots of the cubic derived from \(\cos3\theta=\frac12\), using \(\cos3\theta=4\cos^3\theta-3\cos\theta\).
The PDF gives this hint and the answer \(\frac{19}{16}\).