Simplifying expression with square root of square roots

Question

I have the following expression

 FullSimplify[
 Sqrt[1 + (-1 + a) (-1 + b) + 2 Sqrt[(-1 + a) (-1 + b)]] - Sqrt[
  2 + a (-1 + b) - 2 Sqrt[(-1 + a) (-1 + b)] - b]]

Where $a\ge1$ and $b\ge1$. I know the answer should be $\sqrt{(1-a)(1-b)}$. However, $Mathematica$ doesn't solve this expression this way. How can one resolve this?

Answer 1

Try this:

 expr1 = Sqrt[1 + (-1 + a) (-1 + b) + 2 Sqrt[(-1 + a) (-1 + b)]] - 
      Sqrt[2 + a (-1 + b) - 2 Sqrt[(-1 + a) (-1 + b)] - b];
expr2 = Simplify[expr1 /. {a -> x + 1, b -> y + 1}, {x > 0, y > 0}]

(*  -Sqrt[1 + x y - 2 Sqrt[x y]] + Sqrt[1 + x y + 2 Sqrt[x y]]  *)

Then

expr3=Simplify[expr2 /. x*y -> z^2, z > 0]

Now

expr3 /. z -> (a - 1) (b - 1)

yields

Have fun!