Let's suppose that we're given a holomorphic function
$$f(z) = \dfrac{1}{\sin(z)}$$
And we want to find the coefficient of 20th term in Laurent series. I was trying to apply that $$a_k = \dfrac{1}{2\pi i}\int_{\Gamma} \dfrac{f(z)}{(z-a)^{k+1}}dz$$
for $k = 20$,
$$a_{20} = \dfrac{1}{2\pi i}\int_{\Gamma} \dfrac{f(z)}{z^{21}}dz = \dfrac{1}{2\pi i}\int_{\Gamma} \dfrac{1}{\sin(z)z^{21}}dz$$
Is there a way to evaluate this contour integral? And can we use residue theorem?
Use the formula $g^{(n)} (0)=\frac {n!} {2\pi i} \int_{\Gamma} \frac {g(z)} {z^{n+1}} dz$. Take $n=21$ and $g(z)=\frac z {\sin z}$ which is analytic in side the unit circle.