Mathematica is doing the right thing here it keeps the principal branch alive; if you actually want sqrt(x^2-1) then wrap the whole thing in Assuming[x>1, Simplify[ ... ]] and stop yelling at the CAS for guarding you from your own unstated preconditions.
CASs default to principal branches, so if you want the vanilla sinh(arccosh(x)) = sqrt(x^2 - 1) over x > 1 you have to say so explicitly. I learned that the hard way when a Maple script started spitting negative volumes because a square root wandered across the cut.
Mathematica is doing the right thing here it keeps the principal branch alive; if you actually want sqrt(x^2-1) then wrap the whole thing in
Assuming[x>1, Simplify[ ... ]]and stop yelling at the CAS for guarding you from your own unstated preconditions.CASs default to principal branches, so if you want the vanilla sinh(arccosh(x)) = sqrt(x^2 - 1) over x > 1 you have to say so explicitly. I learned that the hard way when a Maple script started spitting negative volumes because a square root wandered across the cut.