The only way I know how to verify something as an exact differential is when it is in the form $P(x,y) dx + Q(x,y) dy$. The book's definition states that such a differential is exact if there exists a function $g(x,y)$ such that $\frac{\partial g}{\partial x} = P$ and $\frac{\partial g}{\partial y} = Q$.
That's all well and good, but I'm not sure how to express $\cos{z}$ $dz$ in the form $P(x,y) dx + Q(x,y) dy$.
Considering $z=x+iy$ and $cos{z} = \frac{1}{2}(e^{iz}+e^{-iz})$, I know that that $cos{z} = \frac{1}{2}(e^{-y+ix}+e^{y-ix})$, but I'm not sure how $dx$ and $dy$ fit in here or what the rules are for getting them from $dz$. Or even if that question makes sense. Should I even be trying to express $cos{z}$ $dz$ in the form $P(x,y) dx + Q(x,y) dy$?
Please help?