Hi Emanuel, I think that once you do simplify, there would be no need for the quotient rule. That is, since 1/sec(x) = cos(x), then you have: (tan(x) - 1) * cos(x) = sin(x)/cos(x) * cos(x) minus cos(x). This is simply: sin(x) minus cos(x). Differentiating you get: cos(x) plus sin(x).
Perhaps this was a two-part question where they want you to use the quotient rule first (and take a fair amount of time solving), then do it the easy way.