I would like to kindly ask:
does the non-linear equation system
$(I)\ \ \ \ \ \cos(x_1)\cdot \cosh(y_1) = \cos(x_2)\cdot \cosh(y_2)$
$(II)\ \ \ \sin(x_1)\cdot \sinh(y_1) = \sin(x_2)\cdot \sinh(y_2)$
have a non-trivial solution for $x_1,x_2 \in (0,\pi)$ and $y_1,y_2\in\mathbb{R}$?
The trivial solution is: $x_1=x_2$ and $y_1=y_2$.
I tried to square both equations and use $\sin ^2 + \cos^2 = 1$, but it did not lead anywhere.