Proof: $\tan(x)$ is surjective onto $\mathbb R$

To prove $\tan(x)$ defined on $]-π/2;π/2[$ is injective I take the derivative of $\tan(x)$ to get $\sec(x)^2$.

This shows that $\tan(x)$ is monotonic (strictly) increasing which implies it is injective.

However how do I show it is surjective ? That every single real number corresponds to some number in the domain of $\tan(x)$ ?

Thanks in advance

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I can see that a rigorous analytical proof for the surjectivity of $f\colon(-\pi/2,\pi/2)\to\mathbb{R}$, where $f(x)=\tan(x)$, is a way far off. However, I wanted to give a picture of what @achillehui mentioned in a comment, as it is rather beautiful in my opinion:

$\color{white}{Put it in the center!!!!!!!}$enter image description here

Given the picture above, we note that
$$
\tan(\theta)=\frac{x}{1}=x\qquad\text{and}\qquad\tan(-\theta)=-\frac{x}{1}=-x.
$$
Hence, we can “clearly” see that
$$
\lim_{\theta\to\pi/2^-}\tan(\theta)=\infty\qquad\text{and}\qquad\lim_{-\theta\to-\pi/2^+}\tan(-\theta)=-\infty.\qquad\approx\blacksquare
$$