Bài giải:

$\frac{\cos {{60}^{o}}}{1+\sin {{60}^{o}}}=\frac{1}{2}:\left( 1+\frac{\sqrt{3}}{2} \right)=\frac{1}{2}:\frac{\sqrt{3}+2}{2}=\frac{1}{\sqrt{3}+2}$

$\frac{1}{\tan {{30}^{o}}}=1:\frac{\sqrt{3}}{3}=\sqrt{3}$

$\Rightarrow A=\frac{\cos {{60}^{o}}}{1+\sin {{60}^{o}}}+\frac{1}{\tan {{30}^{o}}}=\frac{1}{\sqrt{3}+2}+\sqrt{3}=\frac{1+3+2\sqrt{3}}{\sqrt{3}+2}$

$=\frac{4+2\sqrt{3}}{\sqrt{3}+2}=\frac{2(2+\sqrt{3})}{\sqrt{3}+2}=2$