I am trying to show that $\sin$, $\cos$, $\cosh$ and $\sinh$ only have zeros of multiplicity $1$ in the complex plane. I have found the zeros of each function but I am not sure how to show that they are of order $1$. Can somebody explain?
In complex analysis, the multiplicity of a root is the least $n$ such that the $n$-th derivative of the function is not $0$ at the root (cf. Wikipedia).
If you know that $z$ is a root of $sin$, so $sin(z) = 0$, then to prove it has multiplicity $1$, just show that $sin'(z) = cos(z) \neq 0$.