We are putting here the list of formulas that are the subject of the article mentioned in my older post.
The formula numbering is as in the original article. I have added some new formulas, marked with $?^{CP}$.
$$\prod_{k=0}^{n-1}\sin \frac{\alpha+2k\pi}{2n}=\frac{1}{2^{n-1}}\sin \frac{\alpha}{2}\tag{4}$$
$$\prod_{k=0}^{n-1}\sin \frac{\alpha +k\pi}{n}=\frac{\sin \alpha}{2^{n-1}} \tag{$4^{CP}$}$$
In (4) we put $2\alpha$ instead of $\alpha$.
$$\prod_{k=0}^{n-1}\cos \frac{\alpha +2k\pi}{n}=\frac{1}{2^{n-1}}\left [ \cos \frac{n\pi}{2}-(-1)^n\cos \alpha\right ] \tag{5}$$
$$Interesting\; to\; know \;too\;:\;\;\prod_{k=0}^{n-1}\sin\frac{\alpha +2k\pi}{n}=\fbox{??} \tag{$5^{CP}$}$$
$$\sum_{k=0}^{n-1}\cot^2\frac{\alpha+2k\pi}{2n}=\frac{n(2n-1)+n\cos \alpha}{1-\cos \alpha}\;\;,\;\;\alpha \neq 2m\pi \tag{6}$$
$$\sum_{k=0}^{n-1} \tan^2\frac{\alpha +2k\pi}{2n}=\frac{n(2n-1)+(-1)^n\cdot n\cos\alpha}{1-(-1)^n\cos \alpha}\;\;,\;\;\alpha \neq m\pi \tag{7}$$
$$\sum_{k=0}^{n-1}\cot \frac{\alpha +2k\pi}{2n}=n\cot \frac{\alpha}{2} \tag{$8^{CP}$}$$
$$\sum_{k=0}^{n-1}\cot \left (x+\frac{k\pi}{n} \right )=n\cot nx\;\;\;,\;\;x\in \left (0\;,\;\frac{\pi}{n}\right ) \tag{8}$$
In ($8^{CP}$) we put $\alpha=2nx$.
$$\sum_{k=0}^{n-1}\tan\frac{\alpha+2k\pi}{2n}=\begin{cases}n\tan\frac{\alpha}{2}\;\;,\;n-odd\\-n\cot \frac{\alpha}{2}\;\;,\;n-even\end{cases} \tag{10}$$
$$\sum_{k=0}^{n-1}\tan \left (x+\frac{k\pi}{n}\right )=\begin{cases}n\tan nx\;\;,\;n-odd\\-n\cot nx\;\;,\;n-even\end{cases} \tag{11,12}$$
$$\square\;\;\square\;\;\square$$
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