Can i use trigger ability in a showdown?

Yes — triggered abilities aren't "activated," so they work during a showdown. A trigger fires automatically when its condition is met and goes on the chain as a Pending Item, and Rule 383.3.c allows that during Closed or Open states on any player's turn, showdown or not.

The key distinction is what the ability is. If it starts with "When…", "At…", or "The Nth time…", it's a triggered ability and it triggers during a showdown just fine. If it's written as [Cost]: [Effect], that's an activated ability, and by default it can only be used on your turn, in a Neutral Open state, out of showdown (Rule 381) — so it can't be used in a showdown unless the card grants it Action or Reaction.

One caveat: if a spell you want to play in response has to be Reaction once the trigger starts a chain — a trigger that opens a chain puts the state in Showdown Closed, where only Reaction-speed plays are legal.