Prove that it is an identity sin (A+B) * cos(A-B) = (cos A)^2 - (sin B)^2 Soluti

Question

Prove that it is an identity sin (A+B) * cos(A-B) = (cos A)^2 - (sin B)^2

Explanation / Answer

cos (x + y) = cos x * cos y - sin x * sin y and cos (x - y) = cos x * cos y + sin x * sin y => cos (a + b) * cos (a - b) = (cos a * cos b - sin a * sin b) * (cos a * cos b + sin a * sin b) If x = cos a * cos b and y = sin a * sin b then (cos a * cos b - sin a * sin b) * (cos a * cos b + sin a * sin b) = (x - y) * (x + y) = x² - y² (cos a * cos b - sin a * sin b) * (cos a * cos b + sin a * sin b) = (cos a * cos b)² - (sin a * sin b)² = (cos² a * cos² b) - (sin² a * sin² b) cos² b + sin² b = 1 ===> cos² b = 1 - sin² b and cos² a + sin² a = 1 ===> sin² a = 1 - cos² a => (cos² a * cos² b) - (sin² a * sin² b) = (cos² a * (1 - sin² b)) - ((1 - cos² a) * sin² b) = cos² a - cos² a * sin² b - sin² b + cos² a * sin² b = cos² a - sin² b

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