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

Asked Feb 27 '18 at 07:06

Active Feb 27 '18 at 07:55

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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!

edited Feb 27 '18 at 07:55

asked Feb 27 '18 at 07:06

SHORE SHEN

All their zeroes match..... – ReverseFlowControl Feb 27 '18 at 07:11
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. – ReverseFlowControl Feb 27 '18 at 07:12
@ReverseFlow Thank you mate, but I searched and couldn't find the proof you mentioned. Would you be so kind to be more specific?? – SHORE SHEN Feb 27 '18 at 07:49
@user5713492 thanks, updated~~ – SHORE SHEN Feb 27 '18 at 07:55
See: https://math.stackexchange.com/questions/8337/different-methods-to-compute-sum-limits-k-1-infty-frac1k2-basel-pro – ReverseFlowControl Feb 27 '18 at 08:54
A proof of the Weierstrass product for the sine function over the real line can be found at page 33 of my notes. The key idea is to exploit the factorization of Chebyshev polynomials of the second kind and convexity inequalities for the sine function. – Jack D'Aurizio Feb 27 '18 at 18:20

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