I'm reading the proof below from Rudin's RCA but there are two equations I can't prove. First, how do we get $$|\sum_{S(\theta)} z_k| = |\sum_{S(\theta)} e^{-i\theta} z_k|$$ and how do we get $$\int_{-\pi}^\pi \cos^+(\alpha-\theta)d\theta = 2$$ for every $\alpha$? I would greatly appreciate any proof for these two.
For the first part, notice that $\theta$ is fixed and $\left| e^{-i\theta}\right|=1$
Hence
$$\left| \sum_{S(\theta)}z_k \right|=\left| e^{-i\theta}\right|\left| \sum_{S(\theta)}z_k \right|=\left| \sum_{S(\theta)}e^{-i\theta}z_k \right|$$
For the second part, $\cos \theta$ is $2\pi$ periodic, $\cos ^+(\theta)$ is $2\pi$ periodic too. Hence as long as you integrate over a range of $2\pi$, regardless of where you begin, it gives the same value. Hence $\alpha$ is irrelevant.
$$\int_{-\pi}^\pi \cos^+(\alpha - \theta)\, d\theta = \int_{-\pi}^\pi \cos^+(\theta)\, d \theta = \int_{-\frac{\pi}2}^\frac{\pi}2 \cos \theta \, d \theta = \sin \theta|_{-\frac{\pi}2}^\frac{\pi}2=1-(-1)=2$$