\[ xy=1 \] \[ \text{or, } y=\cfrac{1}{x}=\cfrac{\sqrt3-1}{\sqrt3+1} \] \[ \therefore x+y=\cfrac{\sqrt3+1}{\sqrt3-1}+\cfrac{\sqrt3-1}{\sqrt3+1} \] \[ =\cfrac{(\sqrt3+1)^2+(\sqrt3-1)^2}{(\sqrt3+1)(\sqrt3-1)} \] \[ =\cfrac{2\{(\sqrt3)^2+(1)^2\}}{(\sqrt3)^2-(1)^2} \] \[ =\cfrac{2(3+1)}{3-1} \] \[ =\cfrac{8}{2}=4 \] \[ x-y=\cfrac{\sqrt3+1}{\sqrt3-1}-\cfrac{\sqrt3-1}{\sqrt3+1} \] \[ =\cfrac{(\sqrt3+1)^2-(\sqrt3-1)^2}{(\sqrt3+1)(\sqrt3-1)} \] \[ =\cfrac{4\times \sqrt3\times 1}{(\sqrt3)^2-(1)^2} \] \[ =\cfrac{4\sqrt3}{2} \] \[ =2\sqrt3 \] \[ \cfrac{x^3-y^3}{x^3+y^3} \] \[ =\cfrac{(x-y)^3+3xy(x-y)}{(x+y)^3-3xy(x+y)} \] \[ =\cfrac{(2\sqrt3)^3+3.1.2\sqrt3}{(4)^3-3.1.4} \] \[ =\cfrac{24\sqrt3+6\sqrt3}{64-12} \] \[ =\cfrac{30\sqrt3}{52}=\cfrac{15\sqrt3}{26} \]