Thank you for having patience with my silly question. In my problem, the real "r" was in fact Cosh[x] with x real, which is always >=1 which I neglected to consider. I was asleep at the wheel, and your example woke me up. Thank you. MMA still will not simplify, so now my problem is to find an instance which justifies it.
Clear[c, Coshl];
$Assumptions = {Coshl >= 1, c + Coshl != 0, 1 + c != 0};
FullSimplify[
Sqrt[1/(c + Coshl)] Sqrt[(c + Coshl)/(1 + c)] - Sqrt[1/(1 + c)]]