A Missing Piece: Early Elementary Plane Rotations by Bob Palais
Notices of the AMS February 2014, Doceamus, Web Supplement

This figure from a 2010 Precalculus book shows an alternate approach. The cosine subtraction formula is derived algebraically. Perhaps the benefit of practicing algebraic manipulation is outweighed by the sacrifice of any insight into the structure and relationship to other important relationships. Other derivations will be added below.