Check Laplace's equation: $U_{xx} = e^x\cos y$, $U_{yy} = -e^x\cos y$. So $U_{xx} + U_{yy} = e^x\cos y - e^x\cos y = 0$. Therefore $U$ satisfies Laplace's equation and is harmonic. (It is also the real part of $f(z) = e^z$, which is entire/analytic.)