Computing the value of a Taylor Coefficient

Question

I am having trouble figuring out how to compute the value of the second Taylor coefficient for $e^{\sin(z)}$, with $z=i$. I need to compute the second Taylor coefficient, but I feel after getting help on how exactly to do the first one I can figure it out. This is using $f^k(a)/k!$ of the Cauchy Integral Formula.

Answer 1

To find the second (I assume this means $k=2$, i.e. that there is a "zeroth" coefficient).

This is just

$${f''(i)\over 2!}={1\over 2}\cdot{d\over dz}\bigg|_{z=i}(\cos ze^{\sin z}) = {1\over 2}\cdot e^{\sin i}(\cos^2 i -\sin i)$$