Proving that $\prod_{k=1}^n\left(1+\cot\frac{\pi(4k-1)}{4n}\right)$ is an integer

Question

Let $$P_n=\prod_{k=1}^n\left(1+\cot\frac{\pi(4k-1)}{4n}\right)$$ where $n$ is a natural number and $n\ge3$.

I want to find a simplified expression, from which it is easy to tell $P_n$ is an integer. It is expected to be expressible in the form where $2^{??}$ will appear.

Answer 1

Can't you just exploit the well-known $$ \prod_{k=1}^{m-1}\sin\left(\frac{\pi k}{m}\right)=\frac{2m}{2^m}$$ together with $$ 1+\cot(\theta) = \frac{\sin(\theta)+\cos(\theta)}{\sin(\theta)} = \sqrt{2}\,\frac{\sin\left(\theta+\frac{\pi}{4}\right)}{\sin(\theta)} $$ ? Actually you need just the second identity.