Suppose you have a set of trigonometric functions, which appear unrelated, however they have some conserved properties.
What are the most common operations to find out a relationship between them?
My suggestion is to determine if their $n^{th}$ derivatives are similar to one another. However, I am not sure.
Take for example $\cos(2x)$, $\cos(20x)$, $\sin(2x)+1$:
Evidently, they all intersect at $y=1$, and their slope differs 10-fold.
Then take this example over to a set of nonlinear functions, which are more messy in their wave-behaviour. How can the nonlinear terms that relate these nonlinear functions to one another be found?
de Moivre's formula combined to the Binomial formula are useful to find the expansion of $\cos(nx)$ and $\sin(nx)$.
$$\cos(nx)+i\sin(nx)=(\cos(x)+i\sin(x))^n=\sum_{k=0}^n i^k\binom nk\cos^{n-k}(x)\sin^k(x).$$
You can sometimes eliminate $\cos$ or $\sin$ by means of $\cos^2(x)+\sin^2(x)=1$.