I want to prove that $\tan(x) = x$ has exactly one solution per interval $((n-\frac12)\pi, (n+\frac12))$.
My attempt:
$\tan(x)$ is $\pi$-harmonic, and has a range of $(-\infty, \infty)$ for each interval $(\frac\pi2n, \frac\pi2(n+1)$), and is strictly increasing on each interval. This means that it will cross any linear function exactly once each time.
Have I reached a conclusion here? If so, how can I rewrite the interval to coincide with the one in the problem description?
No, this argument is not sufficient. A function can be strictly increasing and still meet a linear function more than once - for example, $e^x$ meets $x + 2$ twice. The equation $\tan x = 3x$ has three solutions in the interval about $0$.
A hint towards a correct argument: Suppose that we have $\tan(x_1) = x_1$ and $\tan(x_2) = x_2$. Consider the function
$$g(x) = \tan x - x$$
on each interval - this implies $g$ is continuous and differentiable. As a result, the mean value theorem implies that
$$g'(c) = 0$$
for some $x_1 < c < x_2$. But compute $g'$; it doesn't have very many zeros.