I'm trying to prove that
$|\cot (x+iy)|^2=\frac{\cos^2x+\sinh^2y}{\sin^2x+\sinh^2y}$
for $x+iy \in \mathbb{C} \setminus \pi\mathbb{Z}$.
I've tried to use the identities $\cos(iy)=\cosh y$ and $\sin(iy)=i\sinh y$, but I obtain
$|\cot (x+iy)|^2=\frac{\cos^2(x) \cosh^2 y+\sin^2(x) \sinh^2y}{\sin^2(x)\cosh^2 y+\cos^2(x)\sinh^2y}$
What did I miss?