Under what conditions is $||x|-|y|| = |x-y|$ true?

Question

Given that $x,y \in \mathbb{R}$, when is the inverse triangle inequality an equality?

$$||x|-|y|| = |x-y|$$

Answer 1

Suppose they're equal. This is equivalent to their squares being equal (as both numbers are nonnegative): $$ x^2-2|xy|+y^2=x^2-2xy+y^2 $$ This is equivalent to $$ |xy|=xy $$

Answer 2

Only when $x$ and $y$ have the same sign. This is obvious by considering the four possible cases.