, 2024

Solution:

We manipulate the given equation as follows:

$$\begin{aligned} \tan (x+y) &= \tan x + 2 \tan y \\ \frac{\tan x + \tan y}{1 - \tan x \tan y} &= \tan x + 2 \tan y \\ \tan x + \tan y &= (\tan x + 2 \tan y) - \tan x \tan y (\tan x + 2 \tan y) \\ \tan x \tan y (\tan x + 2 \tan y) &= \tan y \\ \tan x \tan y \tan (x+y) &= \tan y \\ \tan y (\tan x \tan (x+y) - 1) &= 0 \end{aligned}$$

Thus, the graph of $\tan (x+y)=\tan x+2 \tan y$ is the union of
- the graph of $\tan y=0$, which is equivalent to $y=n \pi$ for some $n \in \mathbb{Z}$; and
- the graph of $\tan (x+y)=\cot x$, which is equivalent to $2x+y=\frac{\pi}{2}+n\pi$ for some $n \in \mathbb{Z}$.

Each of the above graphs is a disjoint union of equally spaced parallel lines. Thus, the entire graph partitions the plane into congruent parallelograms. To compute the perimeter, we need to pick two adjacent lines from each bullet point.

We pick $y=0$, $y=\pi$, and $2x+y= \pm \pi / 2$. This is a parallelogram with vertices $( \pm \pi / 4, 0)$, $(-3 \pi / 4, \pi)$, and $(-\pi / 4, \pi)$. This is a parallelogram with side lengths $\pi / 2$ and $\pi \sqrt{5} / 2$, so the perimeter is $\pi(\sqrt{5}+1)$.