EAGLES
Shrine Auditorium, Los Angeles
December 31, 1974

Dan Fogelberg opened. They also invited a few guests to join them.
That’s Randy with the beard in the photos from the show.

Los Angles Times, December 29, 1974
Daily Trojan (USC), January 10, 1975
December 31, 1974, Shrine Auditorium, Los Angeles
Randy (left) with J.D. Souther

“When midnight arrived, the Eagles treated everybody to a bluegrass treatment of ‘Auld Lang Syne,’ released a passelfull of balloons and let loose a gorilla who threw bananas to the audience. Then they invited a few of their friends, including Linda Ronstadt, Jackson Browne and opening act Dan Fogelberg–on stage to join them for ‘Take It Easy.'” (San Pedro News-Pilot, January 4, 1975)
Joe Walsh was also among the guests.

L-R: Randy, Joe Walsh, Dan Fogelberg, Glenn Frey, Don Henley, Linda Ronstadt, Jackson Browne.
News-Pilot (San Pedro, CA) – January 4, 1975
Photo: Daryl Nace (Rivendell Gallery)

Randy performed on New Year’s Eve at least nine times over the course of his career. Here are the other appearances:

Drivin’ Dynamics, Little Moon Lake, Torrington, Wyoming, December 31, 1962
Drivin’ Dynamics, Terry’s Arena, Terrytown NE, December 31, 1964
Pogo, Shrine Hall, Los Angeles, December 31, 1968
Roberts-Meisner Band, Calvary Presbyterian Church, San Francisco, December 31, 1987
Roberts-Meisner Band, Sheraton Harbor Island, San Diego, December 31, 1988
Meisner, Swan & Rich, Lake Of The Torches Casino, Green Bay, WI, December 31, 1996
World Classic Rockers, Hilton Waikoloa Village, Hilo, Hawaii, December 31, 2003
World Classic Rockers, Hilton Waikoloa Village, Hilo, Hawaii, December 31, 2005

EAGLES
Red Rocks Amphitheater, Morrison, CO
August 18-19, 1975

Dan Fogelberg opened for the Eagles.

Denver Post, August 17, 1975

Although not billed, Tom Waits performed at the August 18th show, much to the displeasure of the audience:

Unfortunately Tom was booed off the stage before he could do a whole set, presumably because the Eagles fans were more accustomed to sweet than dour. The headliners were not pleased, and made a big point of emphasizing the identity of the composer of ‘Ol 55’ when the time came.” (Ken Langford, Tom Waits Library)

It’s interesting that Waits would perform at this show since he disliked the Eagles version of his song and said so publicly in early 1975 (read more here). The next year his comments about the band were even more harsh when he told a reporter that their albums were only good for “keeping dust off your turntable.” One can’t help but wonder if getting booed at this concert added to his contempt for the band.

Straight Creek Journal (Boulder), August 14, 1975