The Hilbert transform $\hat{f}$ of a function $f$ is defined by

$$\hat{f}(t)=\frac{1}{\pi} \mathcal{P} \int_{-\infty}^{\infty} \frac{f(\tau)}{t-\tau} d \tau,$$

where $\mathcal{P}$ denotes the Cauchy principal value.

Show that the Hilbert transform of $\frac{\sin t}{t}$ is $\frac{1-\cos t}{t} .$