Show that $2\cos(\frac{\theta}{2})=\cos \theta$ iff $\theta = (4n+2)\pi \pm 2\phi$
You have correctly got $$ \theta = \pm 2\cos^{-1}\bigg(\frac{1-\sqrt{3}}{2}\bigg)+4\pi n$$
Now, note that $$\cos^{-1}\bigg(\frac{1-\sqrt{3}}{2}\bigg)=\pi-\phi$$ So, one finally has $$ \theta = \pm 2(\pi-\phi)+4\pi n,$$ i.e. $$\theta=(4n+2)\pi-2\phi,\ (4n-2)\pi+2\phi$$ where the latter can be written as $$\theta=(4m+2)\pi+2\phi$$ where $m=n-1$.
Therefore, one can write $$\theta=(4n+2)\pi\pm 2\phi$$