Consider the cosine function $f = \cos : \Bbb R \to \Bbb R$.
Is it true that the set of iterates $$\left\{f_n := \cos \circ \dotsb \circ \cos, \; n \text{ times } \mid n \geq 1\right\}$$ is linearly independent over $\Bbb R$ ? That is, I am wondering if, for any $r \geq 1$ and any real numbers $a_k$, we have : $$\sum_{k=1}^r a_k f_k = 0 : \Bbb R \to \Bbb R \implies a_k=0 \;\forall k.$$
I know that this true if we consider the powers of $\cos( \cdot )$, but I don't know how to deal with compositions. What I tried is to take derivative, or induction on the minimal length of linear dependence relation.