$(1-\frac{\sqrt{3}-1}2):(\frac{\sqrt{3}-1}2+2)$
$=\frac{2-\sqrt{3}+1}2:\frac{4+\sqrt{3}-1}2$
$=\frac{3-\sqrt{3}}2·\frac{2}{3+\sqrt{3}}$
$=\frac{3-\sqrt{3}}{3+\sqrt{3}}$
$=\frac{(3-\sqrt{3})^2}{(3-\sqrt{3})(3+\sqrt{3})}$
$=\frac{9+3-6\sqrt{3}}{9-3}=\frac{12-6\sqrt{3}}6=2-\sqrt{3}$