Help algebra

$\begin{align*} \frac{1}{1+\frac{1}{\sqrt{3}+2}} &= \frac{1}{\frac{3+\sqrt{3}}{\sqrt{3}+2}} \\ &= \frac{\sqrt{3}+2}{3+\sqrt{3}} \\ &= \boxed{\frac{3+\sqrt{3}}{6}} \end{align*}$