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This is in fact just about Maxima's simplification.
sage: F = asin(sin(x))
sage: F.simplify() # sends to Maxima, nothing else
x
sage: G = acos(cos(x))
sage: G.simplify()
arccos(cos(x))
This Maxima help thread suggests that, as of a couple years ago, this was known but not resolved. I've opened Maxima bug 2570 for this, though I wouldn't expect immediate response as it may be something somewhat deep in the code that isn't easy to unravel - simplification routines can be notorious for this, I guess.
