$\cos (n \phi)$ in terms of $\cos (\phi)$ Consider the following quantity:
$$\cos(n \phi), \ n \in \mathbb{N}, \ n > 1$$
I know it is possible to alternatively express it as a polynomial of degree $n$, with powers of $\cos (\phi)$ and $\sin (\phi)$.
But is it possible to express $\cos(n \phi)$ in terms of $\cos(\phi)$ only? In other words, is it possible to write
$$\cos(n \phi) = A \cos (\phi + \alpha)$$
for some real constants $A, \alpha$, or as a linear combination
$$\cos(n \phi) = A \cos (\phi + \alpha) + B \cos (\phi + \beta) + \ldots$$
?
My guess: no, because the base frequency of $\cos (\phi)$ is too low, therefore unable to represent faster variations with respect to $\phi$, as it happens in $\cos(n \phi)$.

No, it is not possible. Consider differentiating the equation with respect to $\phi$ four times. The LHS gets multiplied by $n^4$ but the RHS does not - a contradiction.
