\frac{2}{\sqrt2+\sqrt3-1}=\frac{2}{\sqrt2+(\sqrt3-1)}*\frac{\sqrt2-(\sqrt3-1)}{\sqrt2-(\sqrt3-1)}=\frac{2(\sqrt2-\sqrt3+1)}{(\sqrt2)^2-(\sqrt3-1)^2}=
\frac{2(\sqrt2-\sqrt3+1)}{2-(3-2\sqrt3+1)}=\frac{2(\sqrt2-\sqrt3+1)}{2-(4-2\sqrt3)}=\frac{2(\sqrt2-\sqrt3+1)}{2-4+2\sqrt3}=
\frac{2(\sqrt2-\sqrt3+1)}{2\sqrt3-2}=\frac{2(\sqrt2-\sqrt3+1)}{2(\sqrt3-1)}=\frac{\sqrt2-\sqrt3+1}{\sqrt3-1}=\frac{(\sqrt2-\sqrt3+1)(\sqrt3+1)}{(\sqrt3-1)(\sqrt3+1)}=\frac{\sqrt6-3+\sqrt3+\sqrt2-\sqrt3+1}{3-1}=\frac{\sqrt2+\sqrt6-2}{2}