If we allow some $g(x) = \frac{\sin(x)}{x}$ such that $g(0) = 1$ and also some $f(x)$ where:
$$f(x) = \prod_{k=1}^{\infty} 1 - \frac{x^2}{\pi^2k^2}$$
Then it is said f and g are equal. But my question is, what are the specifics besides the fact that:
Both have the same value at $x=0$. Both have the same roots, and an infinite number of them of course.
I feel like those two criteria definitely don’t justify equality, so I would like to know what legitimately justifies it.