Answer
Please see below.
Work Step by Step
Let $y$ be any real number; we can find some real numbers $w$ and $z$ such that $w \le y \le z$. Since the range of $f(x)=\tan x$ on the interval $(- \frac{\pi }{2} , \frac{\pi }{2} )$ is the set of all real numbers $\mathbb{R}$, there must exist real numbers $u$ and $v$ in the interval $(- \frac{\pi }{2} , \frac{\pi }{2} )$ such that $f(u)= \tan u =w$ and $f(v)= \tan v=z$.
Please note that the function $f(x)= \tan x$ is continuous at any point in the interval $(- \frac{\pi }{2} , \frac{\pi }{2} )$, so $f(x)= \tan x$ is continuous on the closed interval $[u, v]$. Thus, according to the Intermediate Value Theorem, there must exist some $x$ in $[u, v]$, within the interval $(- \frac{\pi }{2} , \frac{\pi }{2} )$, such that$$f(x)= \tan x = y.$$
