Why are this all the invariant subspaces from the derivation on $\sin$ and $\cos$?

Question

If we consider the function derivative $\phi:f(x)\to \frac{df(x)}{dx}$, over the linear space $V$ given by the spam from $\langle \cos x, \sin x,\dots \cos (nx), \sin (nx)\rangle$, I want to find all invariant subspaces $(\phi(W)\subseteq W)$, is easy to note that for $k \in \mathbb{N}$ the subspace $V_k=\langle \cos(kx),\sin(kx) \rangle$ is invariant under the derivative. And moreover $V=V_1 \oplus V_2 \oplus \dots \oplus V_n $. But how can I show that these are ALL the invariant subspaces? Can you give me some hint?