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Trigonometric sum

Prove that

    \[\cos \frac{\pi}{7} + \cos \frac{3\pi}{7} + \cos \frac{5\pi}{7}=\frac{1}{2}\]

Solution

Consider the tridiagonal matrix A =\begin{pmatrix} 1&1&0\\ 1&0&1\\ 0&1&0 \end{pmatrix} . Its eigenvalues are \displaystyle 2 \cos \frac{\pi}{7} \; , \; 2 \cos \frac{3\pi}{7} \; , \; 2 \cos \frac{5 \pi}{7}. Hence,

    \[\Tr\left ( A \right ) = 1 = 2 \left ( \cos \frac{\pi}{7} + \cos \frac{3\pi}{7} + \cos \frac{5\pi}{7} \right )\]

and the result follows.

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