Divide the reciprocal of $\frac{5}{6}$ by the reciprocal of the reciprocal of $\frac{5}{6}$.
The reciprocal of $\frac{5}{6}$ is $\frac{6}{5}$.  The reciprocal of the reciprocal  of $\frac{5}{6}$ must then be the reciprocal of $\frac{6}{5}$, which is $\frac{5}{6}$.  Thus, the reciprocal of $\frac{5}{6}$ divided by the reciprocal of the reciprocal of $\frac{5}{6}$ is $\frac{6}{5}\div\frac{5}{6}=\frac{6}{5}\cdot\frac{6}{5}=\boxed{\frac{36}{25}}$.