I am looking for the proof that the Hilbert transform of $\displaystyle\frac{\sin(at)}{at}$ is given by

$$\frac{\sin^2(at/2)}{at/2}.$$

How do we prove this? This is a $\operatorname{sinc}(at)$ function. I could not find a proof of this in any textbook so far. Preferably I want to arrive at this result using the analytic signal whose imaginary part would be the Hilbert transform:

$$f\_a(t) = 2 \int^\infty\_0 FT \Big[ \frac{\sin(at)}{at} \Big] \exp(j2 \pi \nu t) \ d\nu$$

I tried this in a few different ways, but could not get to the answer.