trigonometry - How to find the value of $cosleft (frac{pi }{28} right )-cosleft (frac{3pi }{28} right )+sinleft ( frac{5pi }{28} right )$?

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.