Problem when $x=\cos (a) +i\sin(a),\ y=\cos (b) +i\sin(b),\ z=\cos (c) +i\sin(c),\ x+y+z=0$ Problem: If $$x=\cos (a) +i\sin(a),\ y=\cos (b) +i\sin(b),\ z=\cos (c) +i\sin(c),\ x+y+z=0$$ then which of the following can be true:
1) $\cos 3a + \cos 3b + \cos 3c = 3 \cos (a+b+c)$
2) $1+\cos (a-b) + \cos (b-c) =0$
3) $\cos 2a + \cos 2b +\cos 2c =\sin 2a +\sin 2b +\sin 2c=0$
4) $\cos (a+b)+\cos(b+c)+\cos(c+a)=0$
Try: I tried taking $x=e^{ia},y=e^{ib},z=e^{ic}$ and then i tried expressing each option in euler form 
FOR EXAMPLE:
1) $-3/2 e^{-i a-i b-i c}-3/2 e^{i a+i b+i c}+1/2 e^{-3 i a}+1/2 e^{3 i a}+1/2 e^{-3 i b}+1/2 e^{3 i b}+1/2 e^{-3 i c}+1/2 e^{3 i c}$
2) $1/2 e^{i a-i b}+1/2 e^{i b-i a}+1/2 e^{i b-i c}+1/2 e^{i c-i b}+1$
3) $1/2 e^{-2 i a}+1/2 e^{2 i a}+1/2 e^{-2 i b}+1/2 e^{2 i b}+1/2 e^{-2 i c}+1/2 e^{2 i c}$
4) $1/2 e^{-i a-i b}+1/2 e^{i a+i b}+1/2 e^{-i a-i c}+1/2 e^{i a+i c}+1/2 e^{-i b-i c}+1/2 e^{i b+i c}$
Now after all this i'm stuck!!Please help!! How should i proceed?

A hint:
You have three unit vectors in the plane, summing up to $0$. How does the figure look like when you add them geometrically as you add forces?
