ISL 1988 Problem 55

Suppose $\alpha_i > 0, \beta_i > 0$ for $1 \leq i \leq n, n > 1$ and that $$\sum^n_{i=1} \alpha_i = \sum^n_{i=1} \beta_i = \pi.$$ Prove that $$\sum^n_{i=1} \frac{\cos(\beta_i)}{\sin(\alpha_i)} \leq \sum^n_{i=1} \cot(\alpha_i).$$