Question

cos^2A + cos^2 B - 2cosAcosBcos (A+B)=sin^2 (A+B).

Answer 1

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..

Answer 2

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