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This fits right into what Bowie was trying to achieve in America with his Plastic Soul period of '75.
The year before, Alice Cooper went Hollywood with his appearance on Hollywood Squares (with full knowledge of the double entendre in the show's title), and when "Only Women Bleed" hit #1 in March '75, the success of the concept was proven.
Bowie really had no choice but to follow in Alice's wake. America's taste in popular music was polarizing into even harder guitar-driven rock (the pot-and-beer crowd) and the more danceable sound that eventually morphed into disco (the cocaine set). No doubt Angie played a major role in this decision, as probably did the ballad-driven playset of the highly successful Rolling Stones "Tour Of The Americas".
So, an appearance on Cher's show performing a duet with a medley of soulful hits and girl-group backing vocals was pretty much inevitable.