Is there a $z \in \mathbb{C}$ such that both $\sin z$ and $\cos z$ are zero?

Question

Problem Statement

Is there a complex number $z = x + iy \in \mathbb{C}$ ($x, y \in \mathbb{R}$) that satisfies $$\sin z = 0$$ and $$\cos z = 0?$$

My Try

So I have $$e^{iz} = \cos z + i\sin z = 0,$$ which contradicts $$0 = |e^{iz}| = |\exp(iz)| = |\exp(-y + ix)| = \exp(-y)|\exp(ix)|= \exp(-y) > 0,$$ since $y \in \mathbb{R}$.

Question

Is my proof correct?

Answer 1

But $\sin^2z+\cos^2z=1$ for all $z$