Once , I do a problem $ \sum_{n=1}^{14} \cos\left ( \frac{n^{2}\pi }{14} \right )$
when angle modular by $28$ we've got
$2\left ( \cos\left (\frac{\pi }{14} \right ) - \cos\left (\frac{2\pi }{14} \right ) + \cos\left (\frac{3\pi }{14} \right ) + \cos\left (\frac{4\pi }{14} \right ) - \cos\left (\frac{5\pi }{14} \right ) - \cos\left (\frac{6\pi }{14} \right ) \right ) + 1$
and then using identity $\cos\left ( a \right ) - \cos\left ( b \right )$ and $\cos\left ( a \right ) + \cos\left ( b \right )$
leads to $2\sqrt{2} \left (\cos\left ( \frac{\pi }{28} \right ) -\cos\left ( \frac{3\pi }{28} \right )+\sin\left ( \frac{5\pi }{28} \right )\right ) + 1$
the final answer is $\sqrt{7}$ but I don't know how to compute $\left ( \cos\left ( \frac{\pi }{28} \right ) - \cos\left ( \frac{3\pi }{28} \right )+\sin\left ( \frac{5\pi }{28} \right )\right )$ by hands.
I appreciate for your helps.
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