jmc

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 $\frac{6}{5}÷\frac{5}{6}=\frac{6}{5}\cdot \frac{6}{5}=\boxed{\frac{36}{25}}$.