Prove that:
cos^2A + cos^2B - 2cosAcosBcos(A + B) = sin^2(A + B)
LHS:
cos2 A + cos2 B - 2cosA cosB cos(A+B).
= cos2 A + cos2 B - 2cosA cosB (cosA cosB - sinA sinB)
= cos2 A + cos2 B - 2cos2A cos2B + 2 cosA sinA cosB sinB.
RHS:
sin2(A+B)
= (sin(A+B))2
=(sinA cosB+cosA sinB)2
=sin2A cos2B+ cos2A sin2B + 2sinA cosA sinB cosB.
=(1-cos2A) cos2B + cos2A (1-cos2B) + 2sinA cosA sinB cosB.
=cos2B - cos2A cos2B + cos2A- cos2A cos2B.
=cos2 A + cos2 B - 2cos2A cos2B + 2 cosA sinA cosB sinB.
therefore, LHS = RHS.
HENCE PROVED..
