Finding The Zeros Of $(e^z-1) \sin z \cos z$

Question

$$(e^z-1) \sin z \cos z$$

$(e^z-1): z=2\pi i k$ a zero of order $1$

$(\sin z):z=\pi k$ a zero of order $1$

$(\cos z):z=\frac{\pi}{2}+\pi k$ a zero of order $1$

Are those all the zeros?

Answer 1

You got all the zeros, but you missed the fact that $0$ is a zero of order $2$.

Answer 2

This is just an observation:

$$(e^z-1)\sin z\cos z=\frac{1}{2}(e^z-1)\sin 2z$$

So, $z=\frac{k\pi}{2}$ is a zero of order $1$ $\forall k(\ne0)\in\mathbb{Z}$ and if $k=0$ i.e $z=0$ then it is a zero of order $2$ and $z=2k\pi i$ is a zero of order 1 $\forall k(\ne 0)\in \mathbb{Z}$.