When $\cosh (z)=0$?

$$\cosh(z)=0\Longleftrightarrow$$ $$\frac{e^z+e^{-z}}{2}=0\Longleftrightarrow$$ $$e^z+e^{-z}=0\Longleftrightarrow$$ $$e^{-z}\left(1+e^{2z}\right)=0\Longleftrightarrow$$

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Since $e^z$ is never zero for any $z\in\mathbb{C}$, no solution exists for $e^{-z}=0$:

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$$1+e^{2z}=0\Longleftrightarrow$$ $$e^{2z}=-1\Longleftrightarrow$$ $$2z=i\pi(2n+1)\Longleftrightarrow$$ $$z=\frac{i\pi(2n+1)}{2}$$

With $n\in\mathbb{Z}$

Hint

$$\frac {e^z+e^{-z}}{2}=0 \iff e^{2z}=-1$$

Added

Let $z=x+iy$. Let $w:=e^{2z}$ then $w=e^{2z}=e^{2x+2yi}=e^{2x}(\cos2y+i\sin2y)$.

Now, just compare radius and argument with $-1$.

We have $|w|=e^{2x}=1=|-1|$ so $x=0$.

Lastly, \begin{align} \arg(w)&=2y\\ \arg(-1)&=\pi \end{align}

Then, $2y=\pi+2k\pi\rightarrow y=\frac \pi 2+k\pi$.

So we get the solutions: $z=0+\left(\frac{\pi}2+k\pi\right) i$.