Question: How would you prove $$\dfrac 1{R(q)}-1-R(q)=\dfrac {f(-q^{1/5})}{q^{1/5} f(-q^5)}\tag{1}$$ where $f(-q)=(q;q^5)_\infty$ and $R(q)$ is defined as $$R(q)=q^{1/5}\dfrac {(q;q^5)_\infty (q^4;q^5)_\infty}{(q^2;q^5)_\infty (q^3;q^5)_\infty}\tag{2}$$
I'm not sure what to do. I started with Euler's Pentagonal Theorem $$f(-q)=\sum\limits_{n=-\infty}^{\infty} (-1)^n q^{n(3n+1)/2}\tag3$$ and got the following: $$\begin{align*} & \color{red}{f(-q^{1/5})}=\sum\limits_{n=-\infty}^{\infty} (-1)^n q^{n(3n+1)/10}\tag4\\ & \color{blue}{f(-q^5)}=\sum\limits_{n=-\infty}^{\infty} (-1)^n q^{5n(3n+1)/2}\tag5\end{align*}$$ Thus, the RHS can be expressed as $$\dfrac {f(-q^{1/5})}{q^{1/5} f(-q^5)}=\dfrac {\color{red}{\sum\limits_{n=-\infty}^\infty (-1)^n q^{n(3n+1)/10}}}{q^{1/5}\color{blue}{\sum\limits_{n=-\infty}^\infty (-1)^n q^{5n(3n+1)/2}}}\tag6$$ However, I'm not sure how to further reduce down $(6)$. And "even furthermore," I don't know how to simplify $R(q)$ to get $$R(q)=q^{1/5}\dfrac {(q;q^5)_\infty (q^4;q^5)_\infty}{(q^2;q^5)_\infty (q^3;q^5)_\infty}\tag7$$
I checked some of your questions which deal with the work of Ramanujan (like class invariants $G_{n}, g_{n} $, series for $1/\pi$, Rogers-Ramanujan function $R(q) $ of the current question). A lot of Ramanujan's work has been decoded and proved by Dr. Bruce C. Berndt in his five volume work Ramanujan Notebooks and you can get answers to your queries with more details and references there.
On the other hand I have myself studied some material on Ramanujan from various sources (including the Notebooks mentioned above) and have described the same in my blog posts. Material in the blog is self contained and presented in a systematic fashion so that pre-requisites for a result are established before proving the result. I would advise you to have a look at the archives page and check the posts related to "Modular Equations", "Rogers-Ramanujan function", "Class Invariants", "Lambert series", and "Partitions". It also helps if you are aware of the classical theory of elliptic and theta functions as given by Jacobi (so that you can compare it with Ramanujan's work) and some material on these topics is also available on the blog.
I earlier tried to provide an answer here by copy pasting from the blog, but it appears that there are many interconnected results which need to be established first and a coherent answer with all the details would be a bit long and therefore I suggest you take time to study the blog posts mentioned earlier.