Approximation near singularity of $1/\sin$ I'm looking for an approximation $f(x)$ of $\frac{1}{\sin(x)}$ near the singularity at $x=0$.

Can you propose a function or literature or a key word, which leads me to $f(x)$? $f(x)$ must not have a singularity at $x=0$ and needs to be continous.

This may not the best approximation:
$f(x) = k\left(\sin\left(sign(x) \cdot \left|x\right|^c \cdot \frac{3\pi}{2}\right) + x\right)$

The solution for $\frac{1}{\sin(x)}=f(x)$ and $\left(\frac{1}{\sin(x)}\right)'=f'(x)$ for a fixed $k$ and a variable $c$ is hard to find.
Better proposals are welcome.
