\frac{cos60^0}{1+sin60^0}+\frac{1}{tg30^0}=\frac{0,5}{1+\frac{\sqrt3}{2}}+\frac{1}{\frac{\sqrt3}{3}}=
=\frac{0,5}{\frac{2+\sqrt3}{2}}+\frac{3}{\sqrt3}=\frac{1}{2+\sqrt3}+\frac{3}{\sqrt3}=
=\frac{\sqrt3+3(2+\sqrt3)}{\sqrt3(2+\sqrt3)}=\frac{\sqrt3+6+\3\sqrt3}{2\sqrt3+3}=
=\frac{4\sqrt3+6}{2\sqrt3+3}*\frac{2\sqrt3-3}{2\sqrt3-3}=\frac{24-12\sqrt3+12\sqrt3-18}{12-3}=
=\frac{6}{9}=\frac{2}{3}