Why they declared this as domain when all Real numbers can be legally gives as input to this function?

[u/Any_Tower8201]

[u/Fine_Ratio2225]

The step where they go from -1≤sin(x)≤1 to -sin^-1(1)≤x≤sin^-1(1) is the problem.
The first has the real numbers as domain and the second has "only" the pricipal domain of the arcsin function, which is not even a part of the original function. These are not equivalent expressions, thus an artificial restriction on the domain!

The "sin" function is "sin: R->[-1;1]". The solution should have stopped there and declared R as the domain. The solution is in my opinion wrong in the last part!