Let $f(z)$ be a meromorphic function on the complex plane $\mathbb{C}$.
(1). If $f(z)$ has a finite number of poles and $\infty$ is not an essential singularity, then $f(z)$ is a rational function.
(2). If $f(z)$ has at most a finite number of poles and has an essential singularity at $z=\infty$(for example, $f(z)=\frac{e^z}{1+z^2}$), Great Picard's Theorem says that $f(z)$ takes on all possible complex values on $\{z\in\mathbb{C}\mid |z|>R>0\}$, with at most a single exception.
(3). If $f(z)$ has infinte number of poles， say $\{z_n\}$. Then $z_n\to\infty$ and $\infty$ is not an essential singularity. For example, $f(z)=1/\sin z$, $\tan z$.
My Question: In the case (3), Does $f(z)$ also takes on all possible complex values on $\{z\in\mathbb{C}\mid |z|>R>0\}$, with at most two exceptions($f(z)$ takes $\infty$ at poles). Here $\{z\in\mathbb{C}\mid |z|>R>0\}$ is a neighbourhood of $\infty$.
Can anyone recommend some references?