Trapanese completed his bachelor's degree in Composition at the Manhattan School of Music, and later an M.A. in Music for Visual Media at UCLA, with support from the Henry Mancini fund. Joseph's tutors included Giampaolo Bracali, Bruce Broughton, Paul Chihara, Roger Bourland, Jack Smalley, and Martin Bresnick, and he also engaged in brief studies with Louis Andriessen, Aaron Jay Kernis, Julia Wolfe, Mark Snow, and Ira Newborn. From 2008–2011, Trapanese taught the Electronic Music and Composition curriculum at UCLA's Herb Alpert School of Music.
On October 18, 2011, M83's double-disc album Hurry Up, We're Dreaming released with five tracks arranged by Trapanese. Trapanese's work can be heard on "Intro", "Wait", "Soon, My Friend", "My Tears Are Becoming a Sea", and "Outro".
For 2012's The Bourne Legacy, Trapanese worked with Moby on a new orchestral version of his "Extreme Ways".
Trapanese collaborated with Active Child on the arrangement of "Silhouette" (featuring Ellie Goulding) for his 2013 EP Rapor. In addition to the album, he provided new string and choir arrangements for his performances in Melbourne, London, New York City, Chicago, and Los Angeles.
On February 27, 2015 Kelly Clarkson released her seventh album Piece by Piece which featured an orchestra arranged and conducted by Trapanese recorded at EastWest Studios in Hollywood. He collaborated with producers Greg Kurstin, Jesse Shatkin, and Jason Halbert on the project.
In 2017 he scored the Sony Pictures feature Only the Brave, his third collaboration with Director Joseph Kosinski, and co-scored the 20th Century Fox musical The Greatest Showman alongside Composer John Debney. On that film he produced several of the chart-topping songs from the film written by Oscar-winning duo Justin Paul and Benj Pasek.