<s>[INST] Relations similar to $\sin(\pi/18) + \sin(5\pi/18) = \sin(7\pi/18)$ This I checked numerically (and can be proven analytically easily). I guess there are many similar relations between the numbers 
$$\{ \sin\frac{m\pi}{q}, 1\leq m \leq q \} , $$
where $q$ is some integer. 
Can anyone give some reference? 
 [/INST] I have studied this problem and found two kinds of solutions.
[Solution 1]
$ \cos\left(\frac{\pi k}{3 n}\right)=\cos \left(\frac{\pi (n-k)}{3 n}\right)+\cos \left(\frac{\pi (k+n)}{3 n}\right) $
Example:
$ \cos\left(\frac{\pi }{15}\right)=\cos\left(\frac{4 \pi }{15}\right)+\cos \left(\frac{6 \pi }{15}\right) $
$ \cos \left(\frac{2 \pi }{15}\right)=\cos \left(\frac{3 \pi }{15}\right)+\cos \left(\frac{7 \pi }{15}\right) $
$ \cos \left(\frac{3 \pi }{15}\right)=\cos \left(\frac{5 \pi }{15}\right)+\cos \left(\frac{6 \pi }{15}\right) $
$ \cos \left(\frac{ \pi }{21}\right)=\cos \left(\frac{6 \pi }{21}\right)+\cos \left(\frac{8 \pi }{21}\right) $
$ \cos \left(\frac{2 \pi }{21}\right)=\cos \left(\frac{5 \pi }{21}\right)+\cos \left(\frac{9 \pi }{21}\right) $
$ \cos \left(\frac{3 \pi }{21}\right)=\cos \left(\frac{4 \pi }{21}\right)+\cos \left(\frac{10 \pi }{21}\right) $
$ \cos \left(\frac{ \pi }{27}\right)=\cos \left(\frac{8 \pi }{27}\right)+\cos \left(\frac{10 \pi }{27}\right) $
$ \cos \left(\frac{2 \pi }{27}\right)=\cos \left(\frac{7 \pi }{27}\right)+\cos \left(\frac{11 \pi }{27}\right) $
$ \cos \left(\frac{3 \pi }{27}\right)=\cos \left(\frac{6 \pi }{27}\right)+\cos \left(\frac{12 \pi }{27}\right) $
$ \cos \left(\frac{4 \pi }{27}\right)=\cos \left(\frac{5 \pi }{27}\right)+\cos \left(\frac{13 \pi }{27}\right) $
[Solution 2]
$ \cos \left(\frac{\pi  (6 k-3)}{5 (6 k-3)}\right)=\cos \left(\frac{\pi  (10 k-5)}{3 (10 k-5)}\right)+\cos \left(\frac{(2 \pi ) (6 k-3)}{5 (6 k-3)}\right) $
Example:
$ \cos \left(\frac{ \pi }{5}\right)=\cos \left(\frac{  \pi }{3}\right)+\cos \left(\frac{2 \pi }{5}\right) $
$ \cos \left(\frac{3 \pi }{15}\right)=\cos \left(\frac{ 5 \pi }{15}\right)+\cos \left(\frac{6 \pi }{15}\right) $
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