Question

prove that ..,sin(A+B). sin(A-B) =sin^2A-sin^2B​

Answer 1

Answer:

See the explanation

Step-by-step explanation:

  $\sin (A+B)\sin(A-B)$

$=(\sin A\cos B+\sin B\cos A)(\sin A\cos B-\sin B\cos A)$, using identities

$=(\sin A\cos B)^2-(\sin B\cos A)^2$, using $(a+b)(a-b)=a^2-b^2$

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

Hence proved