Prove that: cos²A + cos²B − 2 cosA cosB cos(A+B) = sin²(A+B)
Question
Prove that:
\[ \cos^2 A+\cos^2 B-2\cos A\cos B\cos(A+B) = \sin^2(A+B) \]
Proof
L.H.S.
\[ = \cos^2 A+\cos^2 B \]
\[ -2\cos A\cos B(\cos A\cos B-\sin A\sin B) \]
\[ = \cos^2 A+\cos^2 B-2\cos^2 A\cos^2 B \]
\[ +2\sin A\sin B\cos A\cos B \]
\[ = \cos^2 A(1-\cos^2 B) \]
\[ +\cos^2 B(1-\cos^2 A) \]
\[ +2\sin A\sin B\cos A\cos B \]
\[ = \cos^2 A\sin^2 B +\sin^2 A\cos^2 B \]
\[ +2\sin A\sin B\cos A\cos B \]
\[ = (\sin A\cos B+\cos A\sin B)^2 \]
\[ = \sin^2(A+B) \]
L.H.S. = R.H.S.
Hence proved.