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If you are referring to this paper, then it is still a bit different from this question as that paper uses combinatorial sieve instead of a more GPY-fashioned modern sieve.
I don't see it a good idea to put $\Gamma$ inside some trig functions as it turns out that you will need to estimate terms like $\exp\Gamma$ in the complex plane.
When $q\ge(\log x)^{N+1}$, it follows from $\varphi(q)\gg q/\log\log q$ that $x/\varphi(q)\ll x/(\log x)^N$. When $q\le(\log x)^{N+1}$, we see that (3)'s error term is $\ll x/(\log x)^N$. Therefore, combining both cases gives us (5).
@TerryTao Such abuse of notations are quite common in early works (e.g Brun, Rademacher, and Buchstab) on the development of combinatorial sieve. It was not until Halberstam & Richert that a good set of notation is introduced.
