Hello,

I am reading about how to express the the sine function as an infinite product. I am able to follow the logic of the argument, but a minor detail bothers me. I just hope that someone could help clarify this for me. Euler conjectured that we can express $\displaystyle sinx$ as an infinite product. So, $\displaystyle sinx=x(x-\pm\pi)(x-\pm2\pi)(x-\pm3\pi)\cdot\cdot\cdot=x(\frac{x}{\pi}-\pm1)(\frac{x}{2\pi}-\pm1)(\frac{x}{3\pi}-\pm3)\cdot\cdot\cdot$. Why it is okay to divide each linear factor by $\displaystyle k\pi$? For a finite product, this would not be true. For example, $\displaystyle (x-2)(x+2)\neq(\frac{x}{2}-1)(\frac{x}{2}+1)$. The two expressions have the same roots, but they are not equal. I guess that for infinite product we are allowed to divide the roots, but I do not know why.

In addition, I see a more common expression for the infinite product for sine is $\displaystyle \prod(1-\frac{x}{k\pi})(1+\frac{x}{k\pi})$ instead of $\displaystyle \prod(\frac{x}{k\pi}-1)(\frac{x}{k\pi}+1)$. Why is it okay to swap $\displaystyle 1$ and $\displaystyle \frac{x}{k\pi}$? Again, I guess this has to do with infinite products.