Answer: 994

Small Hint:

Simplify $\cot\alpha+\cot\beta$ using $\alpha+\beta=\pi-\gamma$

Big Hint:

Use the Law of Sines for the resulting sine factors and the Law of Cosines for $\cos\gamma$

Solution:

First, $$\begin{aligned}\cot\alpha+\cot\beta&=\frac{\sin(\alpha+\beta)}{\sin\alpha\sin\beta}\\&=\frac{\sin\gamma}{\sin\alpha\sin\beta}.\end{aligned}$$ Hence the desired ratio is $$\frac{\cos\gamma\sin\alpha\sin\beta}{\sin^2\gamma}=\frac{ab\cos\gamma}{c^2},$$ where the Law of Sines was used in the final equality. By the Law of Cosines, $$\begin{aligned}2ab\cos\gamma&=a^2+b^2-c^2\\&=1988c^2.\end{aligned}$$ Therefore the ratio is $\frac{1988}{2}=994.$