This is more or less standard material on Grothendieck and Serre Duality Theory, and there are several references available.

For instance, the stetement that you want is a consequence of the results in Chapter 5 of S.Ishii's book Introduction to Singularities.

  Look in particular at Theorem 5.3.6, Theorem 5.3.8 and Corollary 5.3.9.

This is more or less standard material on Grothendieck and Serre Duality Theory, and there are several references available.

For instance, the statement that you want is a consequence of the results in Chapter 5 of S.Ishii's book Introduction to Singularities. Look in particular at Theorem 5.3.6, Theorem 5.3.8 and Corollary 5.3.9.

Note that the last corollary also provides an answer to your previous question, since it shows that for any normal, $n$-dimensional variety $X$ the sheaf $\omega_X$ is divisorial, namely reflexive of rank one.