(My personal "feel" is that $\sin$ and $\cos$ are first-class citizens, $\tan$ is "1.5th-class," and the rest are second-class; I'm sure there are others who feel the same.)
Main question(s): From a purely high-school-geometric/"ninth-century-geometer's" standpoint, is there any reason why this should be so? Given the usual elementary knowledge of triangles when one is first introduced to these functions, I think it appears pretty arbitrary. How should I convince a high school student that $\sin$, $\cos$, and $\tan$, instead of their reciprocals, should be our main objects of study? How did history decide on their superiority?
Of course, with real analysis goggles, things look quite a bit different: $\sin$ and $\cos$ are the only ones that are continuous everywhere; $f'' = -f$ characterizes all their linear combinations; they have much nicer series representations; etc. But I suspect this is all hindsight.
(I don't pretend to know enough about complex analysis, but I suspect even more nice things happen there with $\sin$ and $\cos$, and even more ugly things happen with the other four. In any case, I doubt history chose $\sin$ and $\cos$ to be first-class citizens because of their complex properties.)
Secondary questions: Is there any reason why $\sin$ is the "main" function and $\cos$ is "only" its complement, or is this arbitrary as well? Is there any reason why $\tan$ is preferred to $\cot$?