cos (A+B)

問:
Prove cos^2 A + cos^2 (A+B) - 2 cos A cos B cos(A+B) = sin^2 B


解:
Start with LHS. Noting that, in the last term,
　2 cos A cos B = cos(A+B) + cos(A-B) ,
we get LHS = cos^2 A - cos(A+B) cos(A-B) .
But the second team
　cos(A+B) cos(A-B) = 0.5(cos 2A + sin 2B) = cos^2 A - sin^2 B .
Hence, LHS = RHS .


(或:
... But the second team
　cos(A+B) cos(A-B) = cos^2 A cos^2 B - sin^2 A sin^2 B = cos^2 A - sin^2 B
since cos^2 B = 1 - sin^2 B , and sin^2 A = 1 - cos^2 A .
Hence, LHS = RHS . )