a_n \cos nx = \left< f, \frac{\cos nx}{\sqrt{2 \pi}} \right> \frac{\cos nx}{\sqrt{2\pi}} = \frac{\cos nx}{\sqrt{2 \pi}} \int_{-\pi}^\pi \frac{f \cos nx}{\sqrt{2\pi}} dx
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\Leftrightarrow a_n = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) \cos (nx) dx