I'm looking at the solution manual and I have no idea how they convert this.
$ k \cos {3\theta} = k [4\cos^{3} {\theta} - 3 \cos{\theta}] = k[\alpha P_{3}(\cos\theta) + \beta P_{1}\cos{\theta}] $
(I know that any 3rd order polynomial can be express as a linear combination of the first for Legendre polynomials; in this case, since the polynomial is odd, I only need P1 and P3)