<s>[INST] Solving $\cos x + \cos 2x - \cos 3x = 1$ with the substitution $z = \cos x + i \sin x$ I need to solve 
$$\cos x+\cos 2x-\cos 3x=1$$
using the substitution$$z= \cos x + i \sin x $$
I fiddled around with the first equation using the double angle formula and addition formula to get 
$$\cos^2 x+4 \sin^2x\cos x-\sin^2 x=1$$ 
which gets me pretty close to something into which I can substitute $z$, because $$z^2= \cos^2 x-\sin^2 x+2i\sin x\cos x$$ 
I have no idea where to go from there.
 [/INST] Hint:
$$\cos x+\cos2x=2\cos\dfrac{3x}2\cos\dfrac x2$$
$$1+\cos3x=2\cos^2\dfrac{3x}2$$
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