i am having some difficult to expand e^z - 1 in an infinite product. I was using the identity e^z-1 = 2ie^(z/2)sin(z/2i) but i couldn't really figure it out.

also, similar question im trying to prove an identity

1/(e^z-1) = 1/z - 1/2 + 2z ∑(n=1 to infinite) 1/(z^2+4π^2n^2) by taking logarithmic derivatives in e^z - 1

I do not know what to do after... plz help me