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I am trying to prove this formula:

$\prod_{k=1}^{n-1} \sin (\frac{\pi k}{2n}) = \frac{\sqrt{n}}{2^{n-1}}$

I've tried few approaches:

  1. Taking log on it and transforming product to sum of logs.

  2. Coupling $\sin \frac{\pi k}{2n} \sin \frac{\pi (n-k)}{2n}=\frac{1}{2} \sin \frac{\pi k}{n}$

  3. Proof by induction (worked, but i need proof with complex numbers)

  4. Some weird approaches

But nothing worked for me(except induction). I know about sine product formula, but we are just learning complex numbers, i don't want to go that hard. I need to prove it using complex numbers, like taking Im from this expression or something else.

B1Z0N
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    Have you tried substituting $$\sin x=\frac{e^{ix}-e^{-ix}}{2i}$$ – Yuriy S Oct 28 '18 at 20:35
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    You should be able to get some ideas here: https://math.stackexchange.com/questions/324426/finding-prod-n-1999-sin-left-fracn-pi1999-right, https://math.stackexchange.com/questions/8385/prove-that-prod-k-1n-1-sin-frack-pin-fracn2n-1 – Hans Lundmark Oct 29 '18 at 07:08

0 Answers0