Simplifying expression with square root of square roots

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?

• Reduce[{Sqrt[1 + (-1 + a) (-1 + b) + 2 Sqrt[(-1 + a) (-1 + b)]] - Sqrt[2 + a (-1 + b) - 2 Sqrt[(-1 + a) (-1 + b)] - b] == Sqrt[(a - 1) (b - 1)], a > 1, b > 1}] says this is True only when a == (3 + b)/(-1 + b)
– kglr
Feb 19 '19 at 9:28
• They don't seem to match at, for example, a -> 1.23, b -> 2.34. Feb 19 '19 at 9:28

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!