$ \cos (x+a) + \cos( X+b) + \cos( X+c ) = 0$ find possible values of $(c-a) $

Question

Let a, b, c satisfy $ 0 \lt a \lt b \lt c \lt 2 pi $. If $\cos (x+a) + \cos( x+b) + \cos( x+c ) = 0 $ find possible values of (c-a). Closest I can get is $ \tan x = \frac{\cos a + \cos b + \cos c }{\sin a + sin b + sin c } $

Answer 1

Hint: $$\cos(x+a)+\cos(x+c)=-\cos(x+b)$$ and $$\cos(x+a)+\cos(x+c)=2 \cos \left(\frac{a}{2}-\frac{c}{2}\right) \cos \left(\frac{a}{2}+\frac{c}{2}+x\right)$$