Is there a way to fit an even function using only odd functions? I was wondering if there is a way to make an infinite series of odd functions equal to an even function. For example, I would like to know if the next equation is valid
$$\cos(x)=\beta_1x+\beta_2x^3+\beta_3x^5+\beta_4x^7+\cdots$$
Is there any proof that says that is possible?

As Daniel Fischer says, any linear combination of odd functions is still odd.  So if $\cos (2\pi )=1$ is approximated well, your approximation will satisfy $\cos (-2\pi)\approx -1$, which is rather far from $\cos (-2\pi )=1$  This does not strike me as a good approximation.
