In order to earn this trophy, you must have purchased the DLC Future Sound song pack.
To obtain this trophy, you must go to the Rhythm Game menu tracklist, press the X button to open the song playing menu, and press the direction pad over until you see the Performance Video menu (PV) and press X again to confirm playing the video for all songs with the Future Sound label which is located near the Completion percentage. You must watch the PV all the way through to the end for it to count towards the total. If you stop the PV at any point and exit out to the tracklist menu, you must go back to the PV menu and rewatch the entire PV from start to finish to count towards the total. After you've watched the PVs in their entirety for 39 unique Future Sound songs, the trophy should pop at the end of the 39th PV.
It's highly recommended to also take a screenshot while watching through all the PVs, which you can do by pressing Triangle to take a screenshot at any point during the PV, since you can earn another trophy for taking at least one screenshot for 39 unique PVs in Future Sound songs.
WARNING: Do NOT use the Playlist menu to watch the PVs, or it WILL NOT count towards the total. You must watch the PVs from the PV menu in the Rhythm Game tracklist for it to count.