No doubt that $x=\cos\theta$ is a meaningful substitution for the Legendre polynomials. The functions $P_n(\cos\theta)$ were already considered by Legendre in the [spherical harmonic expansion of the Newton potential][1].
However, the substitution is not so happy as for the Chebyshev polynomials. Here below is what you get from the generating function of the Legendre's polynomials -not bad after all, but there may be something better.
$$P_n(\cos\theta)=4^{-n}\sum_{k=0}^n{2k \choose k}{2n-2k\choose n-k}\cos\big((2n-k)\theta\big).$$
  

[1]:https://en.wikipedia.org/wiki/Legendre_polynomials#Expanding_a_1/r_potential