sin(4x) = 2sin(2x)cos(2x) = 2(2sin(x)cos(x))(2cos^2(x)-1) = 4sin(x)cos(x)(2cos^2(x)-1) = 4sin(x)cos(x)(2-2sin^2(x)) = 8cos(x)sin(x) - 8sin^3(x)cos(x) = 4(2cos(x)sin(x) - 2sin^3(x)cos(x))
Теперь возведем это в квадрат:
(sin(4x))^2 = 16(sin(2x))^2 cos^2(2x) = 16(2sin(x)cos(x))^2 cos^2(2x) = 16(4sin(x)cos(x)^3) (2cos^2(x)-1)^2 = 16(4sin(x)cos(x))^3 (4cos^4(x) - 4cos^2(x) + 1) = 16sin^3(x)cos^3(x) (4-4cos^2(x) + 1) = 16sin^3(x)cos^3(x) (5-4cos^2(x)) = 80sin^3(x)cos^3(x) - 64sin^3(x)cos^5(x)
Итак, sin^2(4x) = 80sin^3(x)cos^3(x) - 64sin^3(x)cos^5(x)
sin(4x) = 2sin(2x)cos(2x) = 2(2sin(x)cos(x))(2cos^2(x)-1) = 4sin(x)cos(x)(2cos^2(x)-1) = 4sin(x)cos(x)(2-2sin^2(x)) = 8cos(x)sin(x) - 8sin^3(x)cos(x) = 4(2cos(x)sin(x) - 2sin^3(x)cos(x))
Теперь возведем это в квадрат:
(sin(4x))^2 = 16(sin(2x))^2 cos^2(2x) = 16(2sin(x)cos(x))^2 cos^2(2x) = 16(4sin(x)cos(x)^3) (2cos^2(x)-1)^2 = 16(4sin(x)cos(x))^3 (4cos^4(x) - 4cos^2(x) + 1) = 16sin^3(x)cos^3(x) (4-4cos^2(x) + 1) = 16sin^3(x)cos^3(x) (5-4cos^2(x)) = 80sin^3(x)cos^3(x) - 64sin^3(x)cos^5(x)
Итак, sin^2(4x) = 80sin^3(x)cos^3(x) - 64sin^3(x)cos^5(x)