Complex analysis residue problem. $\oint_\gamma \frac{\sin^2(z)}{(z-1)^2} \,dz$

Question

I am studying for my math exam.

Here is a problem that has stumped me for quite a while now...

$\displaystyle \oint_\gamma \frac{\sin^2(z)}{(z-1)^2} \,dz$

Where $\gamma$ is the circle of radius one centered at $z=1$. I know this problem is asking me to find the residue, but the only techiniques I know for doing this is by expanding $\displaystyle \sin(z)=z-\frac{z^3}{3!}+...$ and maybe by expanding $\displaystyle \frac{1}{(1-z)^2}=(1+z+z^2+...)^2$

Answer 1

Let $f(z):= \sin^2(z)$. Then $\displaystyle \oint_\gamma \frac{\sin^2(z)}{(z-1)^2} \,dz=\oint_\gamma \frac{f(z)}{(z-1)^2} \,dz = 2 \pi f'(1)$, by Cauchy's integral formula.