Show that the ring Q[sin,cos] (say as a subring of the ring of functions on R) is spanned by the functions sin nx and cos nx as n varies.

(This one's from somthng2remembr. There's an unenlightening trivial solution using obscure trig identities, and then a more interesting conceptual solution.)

It seems, though, that I can still harness the field extensions to C, as long as I convince you that the transition matrices between sin(nx) and sin^n(x) are rational? This is easy: cos(nx) = ((cos x + i sin x)^n + (cos x - i sin x)^n)/2, which is clearly rational. But this just unpacks to what I assume is your unenlightening solution?

So right, what you need to use here is something like that the rank of a linear map doesn't change when you extend scalars (it's a little more subtle than what you say though: you need to think about all monomials in sin and cos). But you have to be very careful when thinking about this because it is not a priori clear that extending scalars in this case is the same as tensoring up to C--there could be relations between sin and cos that are not defined over Q.