Why Simplify[1/Sqrt[a] == Sqrt[1/a] doesn't give a "true" output in Mathematica?

Question

In Mathematica, Simplify[1/Sqrt[a] == Sqrt[1/a]] gives Sqrt[1/a] == 1/Sqrt[a]. I don't really see why this doesn't give a true output when, FullSimplify gives the desired result.

This probably is a very dumb question, but I just don't see it.

Answer 1

Because if a<0 then sqrt can have complex output values.

For example

sqrt(1/-1)=i

while

1/sqrt(-1)=-i

See wiki.