Find complex functions on a closed loop

Question

$$\oint \frac{dz}{\cos(z−\pi)(z−3)} \quad \text{over } |z-3|=\frac12.$$ how to calculate by the Cauchy theorem?

Answer 1

Let $f(z)= \cos(z - \pi).$ By Cauchy we have

$$2 \pi i f(3)= \int_{|z-3|=0.5}\frac{f(z)}{z-3} dz.$$

Answer 2

By the Cauchy theorem, the integral is $2\pi i f(3)$, where $f(z)=\dfrac 1{\cos (z-\pi)}$. So $\dfrac{2\pi i}{\cos (3-\pi)}$.