Dirichlet series generating function of Euler totient function

Question

This is a problem from Wilf's Generatingfunctionology, chapter 2. The exercise is to find the Dirichlet series generating function for the totient function $\varphi(n)$, using the facts $$\varphi(p^a) = p^a - p^{a-1}$$ for prime $p$ and $$\sum_{n\geq1}\frac{f(n)}{n^{-s}} = \prod_p \left(1+\frac{f(p)}{p^s} + \frac{f(p^2)}{p^{2s}}+\dots\right)$$ for multiplicative $f$. The solution is $\zeta(s-1)/\zeta(s)$, which I can confirm by multiplying out the terms $$ \left(1+\frac{p-1}{p^s} + \frac{p^2-p}{p^{2s}}+\dots\right)\left(1+\frac1{p^s}+\frac1{p^{2s}}+\dots\right) = \left(1+\frac1{p^{s-1}}+\frac1{p^{2(s-1)}}+\dots\right). $$ However, I'm unable to see how to go the forwards direction and recognize the identity $$ 1+\frac{p-1}{p^s} + \frac{p^2-p}{p^{2s}}+\dots = \frac{1+\frac1{p^{s-1}}+\frac1{p^{2(s-1)}}+\dots}{1+\frac1{p^s}+\frac1{p^{2s}}+\dots}. $$ Could someone help with this? Thanks!