Inifinite product of cos(x)/x?

Question

According to: http://ptrow.com/articles/Infinite_Series_Sept_07.htm

Is there something comparable to this product for$\frac{cos(x)}{x}$?

Answer 1

$$\cos x = \prod_{n=1}^\infty \left(1-\frac{x^2}{(n-\frac12)^2\pi^2} \right).$$

The difference is that the zeros of $\sin$ are at the integral multiples of $\pi,$ but the zeros of $\cos$ are offset by $\pi/2.$

In particular, $0$ is a zero of $\sin,$ but not of $\cos,$ accounting for the "extra" division by $x$ in the formula for $\sin.$ It might be clearer to write these formulas as:

$$\sin x = x \prod_{n=1}^\infty \left(\big(1+\frac{x}{n\pi}\big)\big(1-\frac{x}{n\pi}\big)\right)$$

and

$$\cos x = \prod_{n=1}^\infty \left(\big(1+\frac{x}{(n-\frac12)\pi}\big)\big(1-\frac{x}{(n-\frac12)\pi}\big)\right),$$

which lets you see that these are essentially "factorizations" of $\sin$ and $\cos$ into linear factors based on where their zeros are.

Read about the Weierstrass factorization theorem in complex analysis for more information on this. (Convergence in particular is a critical aspect to look at in general, but the infinite products I gave here converge for all complex $x.)$