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Madalynn Gilbert
Mathematics
27 April, 15:50
Sin (a+b) * sin (a-b) = cos^2b-cos^2a
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Jaden English
27 April, 15:58
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Step-by-step explanation:
To prove sin (a+b) * sin (a-b) = cos^2b-cos^2a
we simplify the left side sin (a+b) * sin (a-b) first
sin (a+b) = sin a cos b + cos a sin b
sin (a-b) = sin a cos b - cos a sin b
sin (a+b) * sin (a-b) = (sin a cos b + cos a sin b) x (sin a cos b - cos a sin b)
sin a cos b ((sin a cos b + cos a sin b) - cos a sin b (sin a cos b + cos a sin b)
open the bracket
sin a cos b (sin a cos b) + sin a cos b (cos a sin b) - cos a sin b (sin a cos b) + cos a sin b (cos a sin b)
sin²a cos²b + sin a cos b cos a sin b - cos a sin b sin a cos b + cos²a sin²b
sin²a cos²b + 0 + cos²a sin²b
sin²a cos²b + cos²a sin²b
sin²a = 1-cos² a sin²b = 1-cos² b
(1-cos² a) cos² b - cos² a (1-cos² b)
= cos² b - cos² a cos² b - cos² a + cos² a cos² b
choose like terms
cos² b - cos² a - cos² a cos² b + cos² a cos² b = cos² b - cos² a + 0
cos² b - cos² a
left hand side equals right hand side
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