Sqrt induces a problem in simplifying

Question

We have an expression such as

 Sqrt[-2 - d^2 + 2 Sqrt[1 + d^2]] Sqrt[-d^2 - 2 (1 + Sqrt[1 + d^2])]

When we use

FullSimplify[Sqrt[-2 - d^2 + 2 Sqrt[1 + d^2]] Sqrt[-d^2 - 2 (1 + Sqrt[1 + d^2])] ]

The result is nothing simplified.

But when we use

FullSimplify[(-2 - d^2 + 2 Sqrt[1 + d^2]) (-d^2 - 2 - 2 Sqrt[1 + d^2])]

in which we firstly expanded the second expression and after that we multiplied them to each other albeit without the Sqrt.

we can see the result is d^4 and Sqrt[d^4] is d^2.

How can we directly obtain the the most simplified yield from the first expression i.e.,

Sqrt[-2 - d^2 + 2 Sqrt[1 + d^2]] Sqrt[-d^2 - 2 (1 + Sqrt[1 + d^2])]

Answer 1

Your purported solution isn't correct. For example:

Sqrt[-2-d^2+2 Sqrt[1+d^2]] Sqrt[-d^2-2 (1+Sqrt[1+d^2])] /. d->.1

-0.01 + 0. I

It helps to add assumptions:

FullSimplify[
    Sqrt[-2-d^2+2 Sqrt[1+d^2]] Sqrt[-d^2-2 (1+Sqrt[1+d^2])],
    d ∈ Reals
]

-d^2

Answer 2

In:

expr = Sqrt[-2 - d^2 + 2 Sqrt[1 + d^2]] Sqrt[-d^2 - 2 (1 + Sqrt[1 + d^2])];
Assuming[d ∈ Reals, Simplify[expr]]

Out:

-d^2