The sin^n x notation is bad. I don't blame the undergrads there. So we learn the following fact: sin^{-1} x = y => x = sin y Being enthusiastic new algebra students, we presume know this must work by applying sine to both sides: sin^{1} sin^{-1} x = sin^{1} y => sin^{0} x = sin y here sin^{0} is zero applications of sine to x => x = sin y Noting we could start at line 2, and apply sin^{-1} to both side also, we have…
Yes, this is definitely a problem. I would just do away with the sin^{-1} notation (as, it seems, many textbooks already do) since we have the perfectly acceptable alternative "arcsin". It's also not a very good notation since it's trying to imply that the sin function has an inverse, but it doesn't. That's why "sin^{-1}(sin(x)) = x" is not even right in general. The inverse only exists on specific subintervals, and…
Of course, the situation is different from many other functions without inverses, because the set of all valid inversions can be trivially generated from one solution. Just put a mirror at pi/2 and -pi/2.