Solution
= Solution
Using the exponential definition of <hyperbolic cosine>, multiply $e^z+e^{-z}=-2$ by the nonzero quantity $e^z$. This yields $(e^z+1)^2=0$, hence $e^z=-1$.
If $z=x+iy$, its <complex modulus> requires $e^x=1$, so $x=0$; its phase then requires $y=(2k+1)\pi$. Thus all complex solutions are
$$
\boxed{z=(2k+1)\pi i,\qquad k\in\mathbb Z.}
$$
There are no real solutions, since real $\cosh x\geq1$.