Equivalency of two radical expressions proof

Question

I know that $$\sqrt{2+2\sqrt{2}}-\sqrt{1+\sqrt{2}}$$ is equivalent to $$\sqrt{\sqrt{2}-1}.$$ However, I do not know how to prove that one is equal to the other and vice versa. The cause that I want to know is that I may stop at the first expression and not simplify further.

Answer 1

$$\sqrt{2+2\sqrt2}-\sqrt{1+\sqrt2}=\sqrt2\sqrt{1+\sqrt2}-\sqrt{1+\sqrt2}=(\sqrt2-1)\sqrt{1+\sqrt{2}}$$

$$ =\dfrac{\sqrt2-1}{\sqrt{\sqrt2-1}}=\sqrt{\sqrt2-1}$$

because $(\sqrt2-1)(\sqrt2+1)=1$.

Answer 2

The LHS is

$$(\sqrt2-1)\sqrt{\sqrt2+1}=\sqrt{\sqrt2-1}$$

by transfering a factor $\sqrt{\sqrt2-1}$.