Answer:
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
