We know from Fourier analysis that we can write
$$f(nt)= \sum_{k=0}^{n} a_kf(t)^k$$
For Fourier basis functions.
Probably the most famous example (special case):
$$n=0: \cos(0t) :\\ 1 = \sin(t)^2+\cos(t)^2$$
Often called the trigonometric unity or "trig-one".
Now to the question.. Can we find other families of functions other than sin, cos and exponential functions where this is also possible?