Fourier series of a fourier series.

Question

let say I have a periodic function I can write $f(t)=a_0+ a_1 \cos(\omega t)+a_2 \cos(2\omega t)...+b_1\sin(\omega t)+b_2\sin(2\omega t)$ that is a fourier series. Is there a link between the coefficients $a_i,b_i$ of $f(t)$ and the $a_i,b_i$ of $\sin(f)(t)$ and $\cos(f)(t)$?