$(\sqrt{3}-1) \sin \theta+(\sqrt{3}+1) \cos \theta=2$

$\Rightarrow \frac{(\sqrt{3}-1)}{2 \sqrt{2}} \sin \theta+\left(\frac{\sqrt{3}+1}{2 \sqrt{2}}\right) \cos \theta=\frac{1}{\sqrt{2}}$

$\Rightarrow \sin \frac{\pi}{12} \sin \theta+\cos \frac{\pi}{12} \cos \theta=\cos \frac{\pi}{4}$

${\Rightarrow \cos \left(\theta-\frac{\pi}{12}\right)=\cos \frac{\pi}{4}} $

${\Rightarrow \theta-\frac{\pi}{12}=2 \mathrm{n} \pi \pm \frac{\pi}{4}} $

$\quad \left[ {\therefore \cos \theta  = \cos \alpha  \Rightarrow \theta  = 2{\rm{n}}\pi  \pm \alpha } \right]$

$\Rightarrow \theta=2 n \pi \pm \frac{\pi}{4}+\frac{\pi}{12},$ $n$ is any integer.