Consider the structure $(\mathbb{R},+,\cdot,\sin)$, where $+$ denotes addition, $\cdot$ denotes multiplication, and $\sin$ denotes the sine function. I conjecture that the equational identities of that structure are generated by the associative and commutative laws of addition and multiplication and the distributive law. Is this conjecture true? In other words, what I am really asking is, does the sine function interact non-trivially with addition and multiplication?
Edit: Also, is there an explicit finite basis for the identities of this structure?
Turning my and Eric's comments above into an answer, combining the formulas $$\cos(2x)=1-2\sin^2(x)$$ and (for example) $$x[\cos^2(2x)+\sin^2(2x)]=x$$ we get $$x[(1-2\sin^2(x))^2+\sin^2(2x)]=x.$$ Expanding out and moving the negative term to the other side yields $$4x\sin^4(x)+x\sin^2(2x)=4x\sin^2(x),$$ which is a nontrivial identity in addition, multiplication, and sine.
I do suspect that there are no nontrivial identities using just addition and sine or just multiplication and sine, however, but I don't immediately see a proof of this. (I've asked a separate question about this.)