In a [recent question](https://math.stackexchange.com/questions/775750/fresh-integral-int-0-infty-frac-ln-cos2-xx2dx-pi), it was stated in a comment, without proof, that

$$ PV \int\_{-\infty}^{\infty} \frac{\tan x}{x}dx = \pi$$

What is the easiest way to prove this? I was able to show that 

$$
PV \int\_{-\infty}^{\infty} \frac{\tan x}{x}dx = -PV\int\_{-\infty}^{\infty} \frac{1}{x \tan x}dx \\
PV \int\_{-\infty}^{\infty} \frac{1}{x \sin x}dx = 0 \\
\int\_{-\infty}^{\infty} \frac{\sin x}{x}dx = \pi
$$

but failed to compute the original integral from this.