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, which is not bad, after all. I'm not sure that there is nothing 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