<s>[INST] Find the value of $\sin\frac{\pi}{7}\sin\frac{2\pi}{7}\sin\frac{3\pi}{7}\sin\frac{\pi}{14}\sin\frac{3\pi}{14}\sin\frac{5\pi}{14}$ 
Find the value of $$\sin\frac{\pi}{7}\sin\frac{2\pi}{7}\sin\frac{3\pi}{7}\sin\frac{\pi}{14}\sin\frac{3\pi}{14}\sin\frac{5\pi}{14}$$

$$\sin\frac{\pi}{7}\sin\frac{2\pi}{7}\sin\frac{3\pi}{7}\sin\frac{\pi}{14}\sin\frac{3\pi}{14}\sin\frac{5\pi}{14}$$
$$=\sin\frac{2\pi}{14}\sin\frac{4\pi}{14}\sin\frac{6\pi}{14}\sin\frac{\pi}{14}\sin\frac{3\pi}{14}\sin\frac{5\pi}{14}$$
$$=\sin\frac{\pi}{14}\sin\frac{2\pi}{14}\sin\frac{3\pi}{14}\sin\frac{4\pi}{14}\sin\frac{5\pi}{14}\sin\frac{6\pi}{14}$$
How can I compute $$\prod\limits_{r=1}^{n}\sin\frac{r\pi}{c}$$
 [/INST] By the basic properties of Chebyshev polynomials the numbers $x_k=\sin(k\pi/14)$, $k=-6,-5,\ldots,5,6, k\neq0$ are the twelve zeros of the polynomial
$$
\frac{U_{13}(x)}x=8192 x^{12}-24576 x^{10}+28160 x^8-15360 x^6+4032 x^4-448 x^2+14.
$$
By the usual relations (aka Vieta relations) of the zeros of a polynomial and its coefficients it follows that
$$
A=\prod_{k=-6..6, k\neq0}x_k=\frac{14}{8192}=\frac{7}{2^{12}}.
$$
Because sine is known to be an odd function, the products $N=\prod_{k=-6}^{-1}x_k$ and $P=\prod_{k=1}^6x_k$ are equal. You are asking about the product
$P$. It is clearly a positive number, so from the equation
$$A=NP=P^2=\frac{7}{2^{12}}
$$
we can solve
$$
P=\frac{\sqrt7}{2^6}.
$$
The method generalizes for other products of sines as long as you take the product of ALL the numbers $\sin (k\pi/(2N))$, $k=1,2,\ldots,N-1$.
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