I find the co-incidence of the identity: $$\sin(A+B)\sin(A-B) = \sin^2 A - \sin^2 B$$ very pleasing. So, I was wondering if there are more of these type of identities.
To make my question precise: Are there more examples of functional equations which are also valid for the identity map?
For example, the identity map and the $\sin$ function satisfies $$(f(A) + f(B))(f(A) - f(B)) = f(A)^2 - f(B)^2.$$
Thanks
For the case of arithmetic functions (i.e. functions whose domain is $\mathbb{N}$) there is the class of completely multiplicative functions which satisfy $f(ab)=f(a)f(b)$ for all positive integers $a$,$b$. Any monomial $x^k$ is an example, but analytic number theory contains examples (Dirichlet characters, for instance) which are far less trivial.