Inequality involving $|\cos(x)|$

Question

If we have: $$\prod_{k=1}^{N}|\cos(\omega x_k)|=1$$ then, how can the following inequality: $$\sum_{k=1}^N\frac{1}{N-1+|\cos(\omega x_k)|}\le1$$ be proven? Thanks:

Answer 1

Hint: $$\prod_{k=1}^{N}|\cos(\omega x_k)|=1 \implies |\cos(\omega x_k)|=1$$ so this is really an equality.