Calculate $\cos(z)/(z^2-\pi^2)$ using Cauchy integral formula on region |z|=4

Question

I want to verify if my reasoning and answer is correct here. Since $\pi$ and $-\pi$ are both contained within the circle centered at 0 with radius 4, we can use the Cauchy integral formula to deal with both singularities which leads to $2 \pi i \cos(\pi)/(2\pi)$. This equals $-i$. However I also know that if you choose $-\pi$ instead you get $i$ instead. So which answer is correct here?

Answer 1

The answer is in fact zero. First note

$$\int_{|z| = 4} \frac{\cos z}{z^2 - \pi^2} \, dz = \frac{1}{2\pi} \int_{|z| = 4} \frac{\cos z}{z - \pi}\, dz - \frac{1}{2\pi} \int_{|z| = 4} \frac{\cos z}{z + \pi}\, dz \tag{*}$$

By the Cauchy integral formula, the first integral on the right evaluates to $i\cos(\pi) = -i$ and the second integral evaluates to $i\cos(-\pi) = -i$. In light of $(*)$, this means that your integral evaluates to $-i - (-i) = 0$.