I have the equations $ - e^y \sin(x) + e^x \cos(y) = 0$ and $e^y \cos(x) - e^x \sin(y) = 0$. I'm stuck as to how to find solutions. I have tried grouping the variables on one side, but to no luck because we are mixing exponentials and $\sin$ and $\cos$. I also tried setting them equal to one another but after grouping variables again I don't see an easy (e.g. trig) formula to use here...
For example $$e^y( \sin(x) + \cos(x)) = e^x (\sin(y) + \cos(y))$$ and then $$\frac{e^x}{\sqrt 2} \csc(x+ \frac\pi4) = \frac{e^y}{\sqrt 2} \csc(y+ \frac\pi4) $$ and then $$ e^{x-y} = \frac{\sin(y + \frac\pi4)}{\sin (x + \frac\pi4)} $$ but I can't see what to do next other than taking natural log of both sides which doesn't seem helpful.