# Linked Questions

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### Evaluation of a product of sines [duplicate]

Possible Duplicate: Prove that $\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$ I am looking for a closed form for this product of sines: \begin{equation} \sin \left(\frac{\pi}{n}\...
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Prove that $$\prod^{n}_{k=1}\sin\left(\frac{k\pi}{2n+1}\right) = \frac{\sqrt{2n+1}}{2^{n}}$$ $\bf{My\; Try::}$ Let $\displaystyle \cos \left(\frac{k\pi}{2n+1}\right)+i\sin \left(\frac{k\pi}{2n+1}\... 1answer 40 views ### Evaluating a product of sines [duplicate] I saw this product from a question, but got deleted. $$\prod_{k=1}^{n-1}2\sin\frac{k\pi}{n}$$ Naturally, I was curious, and evaluated this in mathematica, which suprisingly turns out to be: $$\prod_{k=... 0answers 46 views ### 2^{199}\sin(π/199)\cdots\sin(198π/199) Sine Product Series [duplicate] What will be the value of 2^{199}\sin(\pi/199)\cdots\sin(198\pi/199) ? I could have found in case the functions were cosine but what should i do in case of sine? 63answers 20k views ### Funny identities [closed] Here is a funny exercise$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$(If you prove it don't publish it here please). Do you have similar examples? 4answers 2k views ### How to prove those “curious identities”? How to prove$$ \prod_{k=1}^{n-1} \sin\left(\frac{k\pi}{n}\right) = \frac{n}{2^{n-1}}$$and$$ \prod_{k=1}^{n-1} \cos\left(\frac{k\pi}{n}\right) = \frac{\sin(\pi n/2)}{2^{n-1}}$$5answers 24k views ### Probability that n points on a circle are in one semicircle Choose n points randomly from a circle, how to calculate the probability that all the points are in one semicircle? Any hint is appreciated. 3answers 780 views ### Finding \prod_{n=1}^{999}\sin\left(\frac{n \pi}{1999}\right) [closed] I would appreciate if somebody could help me with the following problem. How can we find the product$$ \prod_{n=1}^{999}\sin\left(\frac{n \pi}{1999}\right)$$4answers 379 views ### How to prove that \sum_{k=1}^{n-1} \frac{1}{1-e^{2 \pi i k/n}} = \frac{n-1}{2}? I came across the fact that$$ \sum_{k=1}^{n-1} \frac{1}{1-e^{2 \pi i k/n}} = \frac{n-1}{2}.$$How can we prove this identity? 3answers 2k views ### Ahlfors “Prove the formula of Gauss” He says: Prove the formula of Gauss:$$ (2\pi)^\frac{n-1}{2} \Gamma(z) = n^{z - \frac{1}{2}}\Gamma(z/n)\Gamma(\frac{z+1}{n})\cdots\Gamma(\frac{z+n-1}{n})$\$ This is an exercise out of Ahlfors. ...

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