If r is real and c is complex, Mathematica says FullSimplify[Sqrt[1/(r+c)]Sqrt[(r+c)/(1+c)]] is not necessarily equal to Sqrt[1/(1+c)]. I cannot find an instance where it is not equal, and I cannot prove that Mathematica is wrong. I can't seem to use FindInstance to find such an instance. It either assumes both are complex, or, specifying Reals, that both are reals.
Clear[r,c];
$Assumptions = {r \[Element] Reals};
FullSimplify[Sqrt[1/(r + c)] Sqrt[(r + c)/(1 + c)] - Sqrt[1/(1 + c)]]
FindInstance[Sqrt[1/(r + c)] Sqrt[(r + c)/(1 + c)] != Sqrt[1/(1 + c)], {r, c}]