This is a follow up of my previous question.
Of course, it makes no sense to discuss the asymptotic behavior of trigonometric functions on the real line, as they oscillates forever and does not even have a limit at infinity.
However, it happens that the limits exist for trigonometric functions with arguments tending to some complex infinity which is not real, i.e. $$\lim_{r\to\infty}f(re^{i\theta})$$ exists for $\theta\ne n\pi$ where $f$ is some trigonometric functions.
For example $$\lim_{r\to\infty}\csc(re^{i\theta})=0$$
Now I become interested in the asymptotic behavior of trigonometric functions near complex infinity. I tried to analyze the functions like $\csc(\frac1z)$ but unfortunately it is not differentiable at $0$ and thus I cannot obtain a Taylor series/Laurent series at $0$.
My question is ($f$ is some trigonometric function)
If we write $$f(re^{i\theta})\sim l+\text{lower order terms}$$ for large $r$, what are the lower order terms exactly?($l$ is the limit value of $f$ when $r\to\infty$, e.g. for $f=\csc$, $l=0$; $f=\cot$, $l=\pm i$)