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

I am 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!

• All their zeroes match..... – ReverseFlow Feb 27 '18 at 7:11
• 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. – ReverseFlow Feb 27 '18 at 7:12
• @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 7:49
• @user5713492 thanks, updated~~ – SHORE SHEN Feb 27 '18 at 7:55
• – ReverseFlow Feb 27 '18 at 8:54