\frac{1}{6+2\sqrt{6}}+\frac{1}{6-2\sqrt{6}}=\frac{6+2\sqrt{6}+6-2\sqrt{6}}{(6+2\sqrt{6})(6-2\sqrt{6})}=\frac{12}{6^2-(2\sqrt{6})^2}=\frac{12}{36-24}=\frac{12}{12}=1