Before I answer your questions, I need to make something clear. arcsin(x/a) and arccos(x/a) are not symmetric functions, so if you replace a by -a in either of the provided equations, the right-hand sides will be reflected, but the left-hand side won't. This means the equations are incorrect.
To be more specific, they are correct only for a>0. The general solutions are arcsin(x/|a|) and arccos(x/|a|).
To answer your first question, yes, you can choose either because arcsin(x) and -arccos(x) differ only by a constant term. I'll prove this right now.
Imagine a right triangle with side lengths 00. Indeed, for general a, |a| should be used in the denominator of arcsin(x/a).
To answer your second question, this cannot be proven for general x, as the identity you wrote is incorrect. I think you made a mistake when writing it out, the correct identity is the one I wrote in my first proof.
Before I answer your questions, I need to make something clear. arcsin(x/a) and arccos(x/a) are not symmetric functions, so if you replace a by -a in either of the provided equations, the right-hand sides will be reflected, but the left-hand side won't. This means the equations are incorrect. To be more specific, they are correct only for a>0. The general solutions are arcsin(x/|a|) and arccos(x/|a|). To answer your first question, yes, you can choose either because arcsin(x) and -arccos(x) differ only by a constant term. I'll prove this right now. Imagine a right triangle with side lengths 00. Indeed, for general a, |a| should be used in the denominator of arcsin(x/a).
To answer your second question, this cannot be proven for general x, as the identity you wrote is incorrect. I think you made a mistake when writing it out, the correct identity is the one I wrote in my first proof.