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I tried for a while to classifiy the singularities of $\frac{1}{z}-\frac{1}{\sin z}$ at the origin, but I am stucked. Is there someone who is able to help me at this point?

user1050421
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1 Answers1

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Use the fact that $\frac z {\sin z} $ and $\frac {\sin z - z} {z^2}$ have removable singularities at zero. (If you are not allowed to use them as "known facts" then prove them). Then your original problem becomes easy.