The Hilbert transform of a function $f(x)$ is defined by

$$\mathcal{H}(f)(y):=\frac{1}{\pi} \mathcal{P} \int_{-\infty}^{+\infty} \frac{f(x)}{y-x} d x$$

Calculate the Hilbert transform of $f(x)=\cos \omega x$, where $\omega$ is a non-zero real constant.