<s>[INST] Evaluating $\int_{|z|=1} \frac{\sin(z^2)}{ \left( \sin(z) \right)^2} dz.$ Consider $z \in \mathbb{C}$ and
$$\int_{|z|=1} \frac{\sin(z^2)}{ \left( \sin(z) \right)^2} dz.$$
How would we integrate this?
 [/INST] $$\text{If}\,\,f(z)=\frac{\sin z^2}{\sin^2z}\,\,,\,\text{then}\,\,Res_{z=0}(f)=\lim_{z\to0}\frac{d}{dz}\left(z^2\frac{\sin z^2}{\sin^2z}\right)=0$$
So the singularity at $\,z=0\,$ is in fact a removable one and thus the integral equals zero.
We can also use power series in a tiny neighbourhood of zero:
$$\frac{\sin z^2}{\sin^2z}=\frac{z^2-\frac{z^6}{3!}+\cdots}{\left(z-\frac{z^3}{3!}+\cdots\right)^2}=\frac{z^2\left(1-\frac{z^4}{3!}+\cdots\right)}{z^2\left(1-\frac{z^2}{3}+\cdots\right)}=\left(1-\frac{z^4}{3!}+\cdots\right)\left(1+\frac{z^4}{3}+\cdots\right)$$
the rightmost parentheses being the power series for $\,\displaystyle{\frac{1}{1-z}\,\,,\,|z|<1}\,$
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