<s>[INST] Trigonometry problem Okay..this one simple problem but I am really stuck and have no idea how to start..
$\cos(a-b)+\cos(b-c)+\cos(c-a)=-\frac32$ we need to prove $\cos(a)+\cos(b)+\cos(c)=\sin(a)+\sin(b)+\sin(c)=0 $ 
 [/INST] $\cos(a−b)+\cos(b−c)+\cos(c−a)=−3/2$
$\cos a \cos b+\sin a \sin b+\cos b \cos c+\sin b \sin c+\cos c \cos a+\sin c \sin a=-3/2 $
$\frac{1}{2}\cos a(\cos a + \cos b +\cos c)-\frac{1}{2}\cos^2a+\frac{1}{2}\sin a(\sin a+\sin b+\sin c) - \frac{1}{2}\sin^2 a+
\frac{1}{2}\cos b(\cos a + \cos b +\cos c)-\frac{1}{2}\cos^2b+\frac{1}{2}\sin b(\sin a+\sin b+\sin c) - \frac{1}{2}\sin^2 b+
\frac{1}{2}\cos c(\cos a + \cos b +\cos c)-\frac{1}{2}\cos^2c+\frac{1}{2}\sin c(\sin a+\sin b+\sin c) - \frac{1}{2}\sin^2 c=-\frac{3}{2}$
$\cos a(\cos a + \cos b +\cos c)+\sin a(\sin a+\sin b+\sin c) - (sin^2 a+\cos^2a)+
\cos b(\cos a + \cos b +\cos c)+\sin b(\sin a+\sin b+\sin c) - (sin^2 b+\cos^2b)+
\cos c(\cos a + \cos b +\cos c)+\sin a(\sin a+\sin b+\sin c) - (sin^2 c+\cos^2c)=-3$
$(\cos a + \cos b +\cos c)^2+(\sin a+\sin b+\sin c)^2=0$
then $\cos a + \cos b +\cos c=0$ and $\sin a+\sin b+\sin c=0$
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