The main identity considered in this article is \[ \begin{aligned} 32(q;q)_\infty^{10} &=9\left(\sum_{n=-\infty}^\infty (-1)^n(2n+1)^3q^{3n(n+1)/2}\right)\left(\sum_{n=-\infty}^\infty (-1)^n(2n+1)q^{n(n+1)/6}\right)\tag{1}\\ &\quad - \left(\sum_{n=-\infty}^\infty (-1)^n(2n+1) q^{3n(n+1)/2}\right)\left(\sum_{n=-\infty}^\infty (-1)^n (2n+1)^3 q^{n(n+1)/6}\right).\end{aligned} \] Three proofs of this identity are presented in this article. The first and second proofs of (1) are the same. The only difference is the proof of Ramanujan’s identity \[ 1+3\sum_{n=1}^\infty \frac{nq^n}{1-q^n}-27\sum_{n=1}^\infty \frac{nq^{9n}}{1-q^{9n}}=\frac{(q^3;q^3)_\infty^{10}}{(q;q)_\infty^3(q^9;q^9)_\infty^3} \] that is used to prove (1). The first proof is interesting and it involves an apparently new identity \[ 2(q;q)_\infty^3\left(1-24\sum_{n=1}^\infty \frac{nq^n}{1-q^n}\right) = \sum_{n=-\infty}^\infty (-1)^n(2n+1)^3q^{n(n+1)/2}. \] The third proof of the main identity involves Ramanujan’s identities for theta functions. It should be noted here that (1) is not entirely new. In fact, M. D. Hirschhorn observed that (1) is equivalent to an identity of L. Winquist [see J. Comb. Theory 6, 56–59 (1969; Zbl 0241.05006)].