Is there one word for the type of smile which shows the following type of emotion:
I’ve got you! Now you can’t escape!
For example, a detective smiles this way on finding an irrefutable piece of evidence that will convict a notorious criminal.
Right off the top of my head, I’d probably go with triumphant.
I would suggest smirk. It means to smile in a smug or self-satisfied manner. It is often used pejoritively, but not necessarily.
If the bad guy was gloating while holding the detective in his clutches, then I'd probably call that a sinister smile. But fortunately, in this case, it's the detective who has the upper hand, and he is flashing a knowing smile.
1. suggesting secret information or knowledge
2. wise, shrewd, or clever
3. deliberate; intentional
There is the collocation predatory smile.