prove that sin(a+b)sin(a-b)=cos²b-cos²a
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Answer:
Identities you need:
sin²A = 1-cos²A
sin(A+B) = sinAcosB - cosAsinB
sin(A-B) = sinAcosB+ cosAsinB
So,
sin(A+B)sin(A-B)
=(sin Acos B - cos Asin B)(sin Acos B + cos Asin B)
=(sin Acos B)² - (cos Asin B)²
=sin² Acos² B - cos² Asin² B
=(1-cos² A)cos² B - cos² A(1-cos² B)
=cos² B - cos² A cos² B - cos² A +cos² A cos² B
=cos² B - cos² A
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