Check if the set is connected or not

Question

I want to check if the following set is connected.

$$\displaystyle{S=\{x\in \mathbb{R}^2 : x_2\cos x_1=\sin x_1\}}$$

This set is equivalent to $\displaystyle{\{x\in \mathbb{R}^2 : x_2=\tan x_1\}}$, isn't it?

The tangens function is not continuous on whole $\mathbb{R}$. Does this mean that we can write this set as a union of two non-empty sets?

Or do we have to do something else?

Answer 1

Let $f:\mathbb R^2\rightarrow \mathbb R$ be given by $f(x_1,x_2)=x_1$. Notice $f(S)= \bigcup\limits_{k \text{ odd integer}} (k\pi, k+2\pi)$.

Since $f$ is continuous and $f(S)$ is not connected we conclude $S$ is not connected.