Answer
Yes; Conservative.
Work Step by Step
For a vector field to be Conservative, $\dfrac{\partial f}{\partial y}=\dfrac{\partial g}{\partial x}$
We have: $f(x,y)=e^{-x} \cos y$ and $g(x,y)=e^{-x} \sin y$
Thus, $\dfrac{\partial f}{\partial y}=-e^{-x} \sin y$ and $\dfrac{\partial g}{\partial x}=-e^{-x} \sin y$
So, $\dfrac{\partial f}{\partial y} = \dfrac{\partial g}{\partial x}$
Therefore, a vector field $F$ is Conservative.
