Is this proof of the infinitude of primes valid?
We must have that $1+2\prod_{p'}p'$ is divisible by some prime $q$, so $1+2\prod_{p'}p' = kq$ for some integer $k$. But then, $$\sin\left(\frac{\pi(1+2\prod_{p'}p')}{q}\right) = \sin \pi k = 0$$ which gives the right-hand equality.