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Before you drop down or glide off the cliff, take some time to scan the area of Arkham Island in Detective Mode from the cliff. You will notice 2x armed henchmen to your left towards Arkham North and a bunch of Arkham guards mostly stationed at each door way in pairs. Drop down and talk to some guards, and head over to the south side of Arkham East where you'll find a Riddler Trophy standing all alone on a platform. Pick up the Trophy and head up towards Botanical Gardens. On the way there or when you leave a building, the Batmobile's alarm goes off. Batman tells Oracle to disable the defense system on the Batmobile so it doesn't seriously injure the people around the Batmobile, especially if the Commissioner is near it. Continue your way into Botanical Gardens and enter in, and to your left is a ventilation shaft, drop down and find your way around to find a Chronicle of Arkham, scan it and leave the same way you enter Botanical Garden. Leaving Botanical Garden find a cemetery to your left towards the the east between Botanical Gardens and Arkham Mansion. Find a wooden shack with it's door opened with a Riddler Trophy standing on the table inside. Pick up the Trophy and head a little northwards in the cemetery.

"The Legacy of this island has been well and truly buried." Find a gave that's been dug up with the coffin exposed, the tomb stone should read Dr.Amadeus Arkham. Scanning the tombstone will unlock the profile of Amadeus Arkham. Now head towards the view of Gotham where you'll notice the tallest building in Gotham skyline reads W-ENT (Wayne Enterprise), zoom in on the building's name and scan it. Scanning it will unlock the profile of Bruce Wayne. After scanning the tower, to your left at the very north side of the cemetery should be a Riddler Trophy hidden behind some grass and in between a few tombstones. Pick up the Trophy and back towards Arkham Mansion.