Simplifying $\prod_{k=3}^{n-1}\cos\left(\frac{\pi}{k}\right)$ I am looking to simplify the following, without the use of capital Pi notation:
$$\prod_{k=3}^{n-1}\cos\left(\frac{\pi}{k}\right)$$
Which is meant to produce the sequence: $\left[1,\ \frac{1}{2},\ \frac{1}{2}\frac{\sqrt{2}}{2},\frac{1}{2}\frac{\sqrt{2}}{2}\frac{1+\sqrt{5}}{4},\frac{1}{2}\frac{\sqrt{2}}{2}\frac{1+\sqrt{5}}{4}\frac{\sqrt{3}}{2}...\right]$.
I have seen identities of a similar structure, such as: $$\prod_{k=1}^{n-1}\sin\left(\frac{k\pi}{n}\right) = \frac{n}{2^{n-1}},\qquad or\qquad \prod_{k=1}^{n-1}\cos\left(\frac{k\pi}{n}\right) = \frac{\sin(\frac{\pi n}{2})}{2^{n-1}}$$
But, I am versed neither in the proofs of these identities, nor the properties of the $\Pi$ notation, so I've had a lot of difficulty trying to simplify this on my own. Dealing with $k$ in the denominator instead of the numerator (like in the two aforementioned identities) is something that I am evidently unequipped to deal with on my own. Thank you to anyone willing to help me out!

Well, you could write
$$ \prod_{k=3}^n \cos(\pi/k) = 2^{2-n} \sum_{signs} \cos\left(\pm \frac{\pi}{3} \pm \frac{\pi}{4} \pm \ldots \pm \frac{\pi}{n}\right) $$
where the sum is over all $2^{n-2}$ possible choices of the $\pm$ signs.
If $n$ is moderately large, those $\pm \pi/3 \pm \ldots \pm \pi/n$ will be rather nasty
rational multiples of $\pi$.  So it's not exactly a "simplification".
For example, if $n=7$ I get
$$ 16^{-1} \left(\cos \left( {\frac {11\,\pi}{420}} \right) +\cos \left( {\frac {13\,
\pi}{140}} \right) +\cos \left( {\frac {27\,\pi}{140}} \right) +\cos
 \left( {\frac {109\,\pi}{420}} \right) \\+\cos \left( {\frac {43\,\pi}{
140}} \right) +\cos \left( {\frac {179\,\pi}{420}} \right) +\cos
 \left( {\frac {59\,\pi}{420}} \right) +\cos \left( {\frac {83\,\pi}{
140}} \right) \\+\cos \left( {\frac {57\,\pi}{140}} \right) +\cos
 \left( {\frac {199\,\pi}{420}} \right) +\cos \left( {\frac {97\,\pi}{
140}} \right) +\cos \left( {\frac {319\,\pi}{420}} \right)\\ +\cos
 \left( {\frac {31\,\pi}{420}} \right) +\cos \left( {\frac {113\,\pi}{
140}} \right) +\cos \left( {\frac {151\,\pi}{420}} \right) +\cos
 \left( {\frac {153\,\pi}{140}} \right) 
\right)$$
