$\sin(x)$ infinite product formula: how did Euler prove it?

Check out this link: Ed Sandifer: How Euler Did It, March 2004

And here: Wikipedia: Basel Problem

The Wikipedia link shows there actually does exist an "elementary" proof of the generalized Basel problem (the evaluation of the Riemann zeta function for positive even integers). Graham, Knuth, and Patashnik, in the text Concrete Mathematics, also give an outline of another elementary proof which is easy to follow, using properties of exponential generating functions and some basic calculus. Although it is not how Euler went about it, the approach certainly would have been within his scope of knowledge.