I'm looking into the Fourier series, and I understand that any function can be written as a sum of an odd and even function. However, I don't understand why any even function can be written as sum of cosines/any odd function can be written as a sum of sines, which all the sources I look into just state as fact. Why does that work? Is it just that since the sum of two even functions is an even function, hence if you add together enough cosines you can approximate any other even function? Edit: the main source that I've been looking at is this https://lpsa.swarthmore.edu/Fourier/Series/DerFS.html which states:
An even function, $x_e(t)$, can be represented as a sum of cosines of various frequencies via the equation $x_e(t)=\sum\limits_{n=0}^{\infty}a_ncos(nω_0t)$.
It then continues with the derivation without elaborating on this idea. Other sources I've looked at for this don't discuss the idea of any function being the sum of even and odd functions, hence they don't bring up this idea.
Cosine is an even function, and the product of an even function and an odd function is an odd function and the integral of an odd function from $-t$ to $t$ is zero and these are the cosine terms.
Similarly, sine is an odd function and the product of an odd function and an even function is odd so the integral of the sine terms is zero.