3.2 Chebyshev angle sums
The Chebyshev roots are located at angles \(\theta _k = (2k + 1)\pi /(2N)\). Unlike the equidistributed roots of unity, these angles carry a phase offset of \(\pi /(2N)\). Nevertheless, discrete orthogonality still forces their trigonometric sums to vanish.
For odd \(m\) with \(0 {\lt} m {\lt} N\),
\begin{equation} \sum _{k=0}^{N-1} \cos \left(m \cdot \frac{(2k + 1)\pi }{2N}\right) = 0. \end{equation}
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Sketch
For odd \(m\), we exploit the symmetry \(\cos (m\pi - \alpha ) = -\cos (\alpha )\) when \(m\) is odd. Pairing terms \(k\) and \(N - k - 1\) yields cancellation. The formalization uses trigonometric reflection identities.
For even \(m\) with \(0 {\lt} m {\lt} N\),
\begin{equation} \sum _{k=0}^{N-1} \cos \left(m \cdot \frac{(2k + 1)\pi }{2N}\right) = 0. \end{equation}
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Sketch
For even \(m = 2\ell \), convert to complex exponentials. The sum becomes
\begin{equation} \Re \left(\mathrm{e}^{i\ell \pi /N} \sum _{k=0}^{N-1} \mathrm{e}^{2\pi i \ell k/N}\right). \end{equation}
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Since \(0 {\lt} \ell {\lt} N\), the geometric sum vanishes by Lemma 2.2.1.