I'm really not sure about this Product-to-sum identity on wiki.
See:
I cannot find this anywhere on the web - does anybody know a reference? Certainly the one wiki gives does not cover it.
I'm guessing it is used for something like $\cos(x)\cos(5x)$ (easier done with the other identities mentioned) to convert into a sum but I'm not sure really. I don't understand the $S=\{1,-1\}^n$ bit.