Addition formulas for Jacobi theta functions, Dedekind's eta function, and Ramanujan's congruences

Previously, we proved an addition formula for the Jacobi theta function, which allows us to recover many important classical theta function identities. Here, we use this addition formula to derive a curious theta function identity, which includes Jacobi's quartic identity and some other important theta function identities as special cases. We give new series expansions for η²(τ), η⁶(τ), η⁸(τ), and η¹⁰(τ), where η(τ) is Dedekind's eta function. The series expansions for η⁶(τ) and η¹⁰(τ) lead to simple proofs of Ramanujan's congruences p(7n+5) ≡ 0 (mod 7) and p(11n+6) ≡ 0 (mod 11), respectively.