Abstract: Recently, Farkas and Kra found some cubic theta function identities from their work on automorphic forms. Shortly thereafter, Farkas and Kopeliovich were able to generalize these to $p$-th order theta function identities using the theory of elliptic functions. We give short, elementary proofs of the cubic identities. We show that the $p$-th order identities follow from more general relations between the coefficients of certain theta functions. Our proof is combinatorial and utilizes certain orthogonal transformations.