Trig2.1

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Prove that \sin(A+B) \sin(A-B)=\sin^2 A-\sin^2 B\, and \cos (A+B) \cos(A-B)=\cos^2 A-\sin^2 B,\,

\sin (A+B) \sin (A-B)=(\sin A \cos B+\cos A \sin B)(\sin A \cos B-\cos A \sin B)=\sin^2 A \cos^2 B-\cos^2 A \sin^2 B\,

\sin^2 A (1-\sin^2 B)-(1-\sin^2 A)\cos^2 B)=\sin^2 A-\sin^2 A \sin^2 B-\cos^2 B +\sin^2 A \cos^2 B=\sin^2 A-\sin^2 B\,

Therefore \sin(A+B) \sin(A-B)=\sin^2 A-\sin^2 B\,

\cos (A+B) \cos (A-B)=(\cos A \cos B-\sin A \sin B)(\cos A \cos B+\sin A \sin B)=\cos^2 A \cos^2 B-\sin^2 A \sin^2 B=\cos^2 A(1-\sin^2 B)-(1-\cos^2 A)\sin^2 B\,

\cos^2 A-\cos^2 A \sin^2 B-\sin^2 B+\cos^2 A \sin^2 B=\cos^2 A-\sin^2 B\,

Therefore \cos (A+B) \cos(A-B)=\cos^2 A-\sin^2 B,\,


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