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I am trying to proof the following equation:

$$\sin (x)= x\prod_{i=1}^\infty \left( 1-\frac{x^2}{n^2\pi^2} \right).$$

I got a proof but there is one point that I do not understand.

There is one step during the proof presented the following equation

$$ 1 \geq \prod_{k=m}^n\left\{ 1-\frac{\sin^2 \frac{x}{2n-1}}{\sin^2 \frac{k\pi}{2n-1}} \right\} $$

I am kind of confusing and do not know how to prove it. If anyone can help me and prove it in detail. Thanks!

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  • $\begingroup$ All their zeroes match..... $\endgroup$ – ReverseFlow Feb 27 '18 at 7:11
  • $\begingroup$ You might want to look up Euler's Original proof for $\sum_{k=1}^{\infty} \frac{1}{k^2}=\frac{\pi^2}{6}$. That will give you some ideas. $\endgroup$ – ReverseFlow Feb 27 '18 at 7:12
  • $\begingroup$ @ReverseFlow Thank you mate, but I searched and couldn't find the proof you mentioned. Would you be so kind to be more specific?? $\endgroup$ – SHORE SHEN Feb 27 '18 at 7:49
  • $\begingroup$ @user5713492 thanks, updated~~ $\endgroup$ – SHORE SHEN Feb 27 '18 at 7:55
  • $\begingroup$ See: math.stackexchange.com/questions/8337/… $\endgroup$ – ReverseFlow Feb 27 '18 at 8:54

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