A remark on modular equations involving Rogers-Ramanujan continued fraction via $5$-dissections

In this paper, we study the $5$-dissections of certain Ramanujan's theta functions, particularly $\psi(q)\psi(q^2), \varphi(-q)$ and $\varphi(-q)\varphi(-q^2)$, and derive an identity for $q(q;q)_{\infty}^6/(q^5;q^5)_{\infty}^6$ in terms of certain products of the Rogers-Ramanujan continued fraction $R(q)$. Using this identity, we give another proof of the modular equation involving $R(q), R(q^2)$ and $R(q^4)$, which was recorded by Ramanujan in his lost notebook, and establish modular equations involving $R(q), R(q^2), R(q^4), R(q^8)$ and $R(q^{16})$.