When looking up how the extremely famous series $$\sin(x)=\sum_{k=0}^\infty(-1)^k\frac{x^{2k+1}}{(2k+1)!}$$ is derived, I found this great explanation by Proof Wiki.
My question is this: the explanation shows clearly how to derive the Maclaurin series for $\sin(x)$ and how it converges for all real arguments, however - as someone new to the intricacies of Maclaurin series - it does not prove that whatever the series converges to at the real number $a$ is $\sin(a)$. Why is this true?