Question

Cos²A+cos²B-2cosAcosBcos(A+B)=sin²(A+B)

Answer 1

Step-by-step explanation:

To prove : $\cos^2A+\cos^2B-2\cosA\cosB\cos(A+B)=\sin^2(A+B)$

Proof :

Taking LHS,

$LHS=\cos^2A+\cos^2B-2\cosA\cosB\cos(A+B)$

$=\cos^2A+\cos^2B-2\cosA\cosB(\cosA\cosB-\sinA\sinB))$

$=\cos^2A+\cos^2B-2\cos^2A\cos^2B+2\cosA\cosB\sinA\sinB$

$=\cos^2A-\cos^2A\cos^2B+\cos^2B-\cos^2A\cos^2B+2\cosA\cosB\sinA\sinB$

$=\cos^2A(1-\cos^2B)+\cos^2B(1-\cos^2A)+2\cosA\cosB\sinA\sinB$

$=\cos^2A\sin^2B+\cos^2B\sin^2A+2\cosA\cosB\sinA\sinB$

$=\sin^(A+B)$

=RHS

Hence proved

#Learn more

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

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