There's apparently a geometric proof. Perhaps see if Khan academy has it and do a comparison
I think using manim would be be interesting to show this visually
Perhaps create two part, geometric and rotation proof
Manga Guide to Linear Algebra has a nice quick explanation of rotations
perhaps make it a 3 part for now and then combine them into one that is a condensed version
Math Girls, in the first few pages showed where double angles came from which was very neat.