Вычислите cos(п/4-a)*sin(3п/4-a) если sin a(1-2sin^2a/2)=1/3

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Дано: sin(a)(1 - 2sin^2(a)/2) = 1/3

Упростим уравнение:
sin(a)(1 - sin^2(a)) = 1/3
sin(a)cos^2(a) = 1/3
cos^2(a)sin(a) = 1/3
cos^2(a)*(2sin(a)cos(a)) = 1/3
2cos^2(a)sin(a)cos(a) = 1/3
2sin(a)cos(a) = 1/3
sin(2a) = 1/3

Теперь найдем cos(π/4 - a) и sin(3π/4 - a), используя полученное значение sin(2a):

cos(π/4 - a) = cos(π/4)cos(a) + sin(π/4)sin(a) = (√2/2)cos(a) + (√2/2)sin(a) = √2cos(a)/2 + √2sin(a)/2 = √2(sin(a) + cos(a))/2

sin(3π/4 - a) = sin(3π/4)cos(a) - cos(3π/4)sin(a) = (-√2/2)cos(a) - (-√2/2)sin(a) = -√2cos(a)/2 + √2sin(a)/2 = √2sin(a)/2 - √2cos(a)/2 = √2(sin(a) - cos(a))/2

Теперь вычислим произведение cos(π/4 - a) и sin(3π/4 - a):

cos(π/4 - a) sin(3π/4 - a) = (√2(sin(a) + cos(a))/2) (√2(sin(a) - cos(a))/2)
= 2(sin(a) + cos(a))(sin(a) - cos(a))/4
= 2(sin^2(a) - cos^2(a))/4
= (2sin^2(a) - 2cos^2(a))/4
= 2(sin(2a))/4
= 1/3

Итак, cos(π/4 - a) * sin(3π/4 - a) = 1/3.