Real numbers inequality

Hint. Given 9 random and distinct real numbers in the interval $(-\pi/2,\pi/2)$ whose size is $\pi$, by the pigeonhole principle there are at least two of them, say $x$ and $y$, such that $$0<x-y<\pi/8.$$ Note that $\arctan(\sqrt{2}-1)=\pi/8$.

Another hint:

That formula of $\frac{m-n}{1+mn}$ resembles $\tan(x-y)$.