Locus of Complex number Collapsing into a single point.

Question

Suppose we have the Locus $$|z-1|=1$$ obviously it is supposed to represent a circle of radius 1 and centered at $(1,0)$. But now if i expand it I get , $$z^{2}+1+2zcos\theta =1$$ which eventually gives us $$|z|(|z|-2cos\theta )=0$$. But we know $Arg(z)$ is nothing bus $2cos\frac{\theta }{2}$ by taking $$z-1=e^{i \theta }$$ which simplifies to $$z = 2cos\frac{\theta }{2}e^{i\frac{\theta }{2}}$$ Now from this we get $\theta$ must be $0,2\pi...$. Which only represents the points of z on real axis and not the other points. Why does this happen?