How would I construct a second degree Legendre Polynomial for $f(x)=cos(x)$ on the interval $[-1,1]$? I am not understanding the explanation in the book. I just want to know how to start.
A second order polynomial can be uniquely defined with three distinct points (in general, an order $n$ polynomial requires $n+1$ points).
What does $\cos x$ look like on $[-1,1]$? Well, $\cos$ is symmetric about the $y$-axis. So let's use $x = -1, 0, 1$ as our interpolating points. Then, we wish to find a polynomial such that $P(-1) = \cos (-1)$, $P(0) = \cos 0$, $P(1) = \cos 1$.
Then, it depends on how you're defining the Legendre polynomials. There is a common definition for the Legendre family of orthogonal polynomials, but I'm not entirely sure this is what you are looking for.