It appears as if Bungie’s upcoming expansion will add new achievements/trophies. According to Planet Destiny, a popular Destiny website that attended E3, The Taken King will be the first expansion to increase the amount of achievements and trophies in Destiny.
The Taken King DOES have new achievements/trophies!
— Planet Destiny (@DestinyNews_net) June 18, 2015
In an interview with Planet Destiny, Luke Smith confirmed that there will indeed be new achievements and trophies with the expansion. Planet Destiny is a reputable website, with popular players and Bungie employees including community manager DeeJ appearing on their podcast. The two previous expansions, Dark Below and House of Wolves, did not add any new achievements/trophies. Bungie has yet to confirm or deny the claim. Stay tuned for more information.