Finding the roots of $\sec^2(x)=0$

Question

I need to find the roots of $\sec^2(x)=0$ in my works.

I know there are no real roots of this equation; are there complex roots?

Answer 1

$\sec(x)=\frac2{e^{ix}+e^{-ix}}$ and since $e^{ix}$ is finite for all $x$, there is no $x\in\mathbb{C}$ that will make $\sec(x)=0$.

Answer 2

$tan(x)$ is monotonously increasing with no point yielding extrema, its derivative $sec^2(x)$ is never zero either for real or complex roots.