Answer: The reciprocals of $\frac{1}{-5}$ and $-\frac{1}{6}$ are $-5$ and $-6$, respectively.
The sum of these reciprocals is $-5 + (-6) = -11$.
Now, we need to find the reciprocal of $-11$, which is $\frac{1}{-11} = \boxed{-\frac{1}{11}}$.
So, the reciprocal of the sum of the reciprocals is $-\frac{1}{11}$
Step-by-step explanation:
