= Solution
{c}
The only possible finite <singularities> are $z_\pm=\pm i\pi/2$, the zeros of $4z^2+\pi^2$. They are simple zeros of the denominator, because its <derivative> $8z_\pm$ is nonzero. They are also simple zeros of the numerator: $\cosh z_\pm=0$ and $\sinh z_\pm=\pm i\ne0$.
The common simple factors cancel, so both points are <removable singularities>, with extension values
$$
\boxed{\lim_{z\to z_\pm}\frac{\cosh z}{4z^2+\pi^2}=\frac{\sinh z_\pm}{8z_\pm}=\frac1{4\pi}.}
$$
There are no other finite <singularities>; after inserting these two values the <function> is <entire>.
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