How do you prove a hilbert transform?

Question

I am stuck with this question below, I need help;

Answer 1

Here is how. First use the trigonometric identity

$$ \sin(at+b) = \sin(at)\cos(b) + \cos(at)\sin(b).$$

to expand the trig. expression you have then we need Hilbert transform of $\sin(at)$ and $\cos(at)$ (assuming $a>0$) which are given respectively by

$$ \cos(as)\quad \rm {and} \quad -\sin(as) $$

I think you can advance now.

Note:

1) I only considered the case $a>0$. For the case $a<0$ Hilbert transform of $\sin(at)$ and $\cos(at)$ will be given by

$$ -\cos(as)\quad \rm {and} \quad \sin(as) $$

2) The Hilbert transform is defined as

$$ H[f(x)]=\frac{1}{\pi} \rm{pv}\, \int_{-\infty}^{\infty}\frac{f(x)}{x-y}dx. $$