A simple expression which Mathematica will not simplify

Question

Sorry for the vague title, but I wasn't sure how to phrase this.

I was using Mathematica to solve a fairly simple algebraic equation, but when I tried to substitute the solution into the original equation as a check, I can't seem to recover my original answer. Minimal example:

eq = a == (1 - b)/Sqrt[1 - 2 b]

sol = Solve[eq, b]

FullSimplify[(1 - b)/Sqrt[1 - 2 b] /. sol, a > 0]

The final line outputs

{(a (a + Sqrt[-1 + a^2]))/Sqrt[-1 + 2 a (a + Sqrt[-1 + a^2])], 
 (a (a - Sqrt[-1 + a^2]))/Sqrt[-1 + 2 a (a - Sqrt[-1 + a^2])]}

even after I've specified that a is positive. So FullSimplify is not enough to make this last line output a. What am I missing?

Answer 1

One way to demonstrate is to 'take care of' of the square root: Rewriting:

eqn = (1 - b)/Sqrt[1 - 2 b];
sol = Solve[eqn == a, b];

Now,

FullSimplify[Sqrt[eqn^2 /. sol], a > 0]

yields:

{a, a}