# The graph of tan(sec(x))

A lot of the trig_function(trig_function(x)) look something like this, with asymptotes that have infinite (?) oscillating (?) lines infinitely approaching them (?).

I guess my question, less crude, is - what is going on here? I understand that there are asymptotes at $\frac{\pi}2$ and $\frac{3\pi}2$, but why are there an infinite number of oscillations approaching those x values? Does this sort of graph have a "title"?

Thanks.

$\lim_{x\rightarrow \frac{\pi}{2}^-}\sec x=+\infty$
In particular, the solutions of $\sec x=\frac{\pi}{2}+k\pi,\ k\in\mathbb{N}$ accumulate on the left of $x_0=\frac{\pi}{2}$. But every time $\sec x=\frac{\pi}{2}+k\pi,\ k\in\mathbb{N}$, $\tan\sec x$ has a vertical asymptote. Which gives you the graphical effect you see on the left of $x_0$.
The same argument works on the right, for every singularity of $\sec x$ (i.e. $\frac{\pi}{2}+k\pi,\ k\in\mathbb{Z}$).