Casting a Trap by paying its alternative cost doesn’t change its mana cost or mana value. The only difference is the cost you actually pay.

A spell or ability destroys a permanent only if that spell or ability specifically contains the word “destroy” in its text. If a spell or ability an opponent controls exiles a noncreature permanent you control, causes you to sacrifice a noncreature permanent, removes all loyalty counters from a planeswalker you control, or causes an ability you control to trigger (and then that ability destroys a permanent you control), Cobra Trap’s alternative cost condition hasn’t been met.

Effects that increase or reduce the cost to cast a Trap will apply to whichever cost you chose to pay.

You may ignore a Trap’s alternative cost condition and simply cast it for its normal mana cost. This is true even if its alternative cost condition has been met.