27 May 2020

 I'm tired, so I might muddle the explanation... but this could be used to create the formulae for cos (a+b) and sin (a+b)

Update: So, basically... cos (a+b) is the horizontal side on the top left white triangle, which is the bottom side of the side of the square minus the top right triangle's horizontal side.

Rewritten, we could say that:

cos (a+b) + sin a sin b = cos a cos b (...and subtracting)

cos (a+b) = cos a cos b - sin a sin b

Similarly, using vertical sides -

left square side = sum of lengths on right square side

sin (a+b) = cos a sin b + sin a cos b

Verifying that each segment length in the figure is the combination of trig functions claimed mostly involves finding the right triangle(s) that use that segment length and computing the trigonometric functions on that triangle.

Now, how did someone think of this figure to begin with? Hmmm... good question. Perhaps a good starting point might have been thinking about what the formula does: converting from trig functions involving an angle that is a sum to a calculation using only the individual angles. Like... if I start with angle (a+b) and break it down into the angles a and b; what can we create from those 3 angles? (It's a conjecture, on my part... and may not have been what the original prover was thinking)