Revealing Skill

A Disguise Trope. The Secret Identity of a character is revealed when the character carelessly exhibits a rare skill that they are known to possess.

The skill in question is not necessarily exclusive to the character, just rare enough to set them apart from the crowd.

The distinct characteristic of this trope is that there is no moral dilemma; the character is just acting carelessly. If, on the other hand, the character must risk drawing attention to themselves in order to achieve an important goal, like getting out of a dangerous situation, it's a Cover-Blowing Superpower. If the character in question is a doctor, it's Inconvenient Hippocratic Oath.

Compare and contrast I Never Said It Was Poison, where the character's identity is revealed when he inadvertently reveals he has specific knowledge only he would have access to.

See also Something Only They Would Say where a particular character trait gives the person's identity away instead of a unique skill. I Am Not Left-Handed is when their level of skill is being hidden due to an intentional handicap.

Examples of Revealing Skill include:

Comic Books

In the backstory of the third Robin (Tim Drake), this is how he learned the secret identities of Batman and Robin (Dick Grayson): by watching news coverage of B&R's escapades, during which Robin performed a complicated gymnastics move—which it had been established could be performed only by orphaned circus artist Dick Grayson.

Film

In The Long Kiss Goodnight, the protagonist is a sweet innocent soccer mom with a memory loss - she hurt her head five years ago, and her entire life before that is forgotten. Shortly after the start of the movie, she's suddenly creepily good with daggers. She says that she must have been a chef before the accident, but it's undeniable that she as well as her family have received the first clue that she was really an assassin.
In Crouching Tiger, Hidden Dragon, Shu Lien suspected that the governor's daughter Jen was more than she claimed to be, and had martial arts training. Shu Lien confirmed these suspicions by deliberately dropping a tea cup—Jen caught it in midair and returned it to the table, without spilling a single drop.
After RoboCop outperforms everyone at the firing range, Lewis notices him twirl his Auto-9 before holstering it. Alex Murphy had been doing that in imitation of a TV hero his son watched.

Literature

In Harry Potter and the Deathly Hallows, Harry is revealed when he uses the Disarming Charm — his signature move — against a Death Eater. This element is ignored in the movie.
In the Inheritance Cycle, Eragon recognizes a Rider he's fighting as Murtagh when he flourishes his sword in a certain way.
In Les Misérables, Valjean makes Javert suspicious when he exhibits a feat of strength similar to one he performed before he became a wanted fugitive.

Oral Tradition, Folklore, Myths and Legends

In most versions of Robin Hood, his identity is revealed when he exhibits Improbable Aiming Skills at an archery contest.
As the myth goes, when Daedalus was in hiding, his identity was finally revealed when he solved the problem of passing a string through a sea shell, showing his rare intellect.

Video Games

The Pokémon Zorua and Zoroark, both possessing an ability which disguises them as another of the trainer's Pokémon, can be recognized when they use moves that their disguise could not possibly learn. Optionally, you could just hit them.
The Mythic Dawn sleeper agents in The Elder Scrolls IV: Oblivion don't hesitate to summon up their distinctive magic armour whenever a fight breaks out.

Western Animation

In one episode of Batman: The Animated Series, Bruce Wayne has to try and avoid this, holding back in a fight against his opponent (who knows he's Batman) in front of his Girl of the Week (who doesn't). She knows that he has some fighting skill, but doesn't want to give himself away by showing too much skill.
In The Legend of Korra, the title character joins a pro-bending team, the Fire Ferrets, as their waterbender. During her first match though, she accidentally reveals herself to be the Avatar when she earthbends to block an opponent's attack.