I need to find the residue of f(z) = 1/(sin(z)) at z=Pi

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I had a similar exercise for z=Pi/2. I found the expansion of f(z) to be 1/z + z/6 + ...

then calculating the residue for a simple pole (?) i found that lim as z-->Pi/2 of (z - Pi/2)*(1/z + z/6 + ...) = 0

I find I get the same residue for z=Pi, but I think this is wrong