\sin x+\sqrt{3}\cos x=1 /^2\\

\sin^2 x+2\sqrt{3}\sin x\cos x+3\cos^2 x=1\\

(\sin^2 x+\cos^2 x)+2\sqrt{3}\sin x\cos x+2\cos^2 x=1\\

2\sqrt{3}\sin x\cos x+2\cos^2 x=0\\

\cos^2x\left(\sqrt{3}\tan x+1\right)=0\\

\cos^2 x=0\,\,\vee\,\,\tan x=-\frac{\sqrt{3}}{3}\\

x_1=\frac{\pi}{2}+k\pi\,\,\vee\,\,x_2=-\frac{\pi}{6}+2k\pi