а) sin(x-π/3) cos(x-π/6) = sin(x)cos(π/3)cos(x)sin(π/6) = sin(x) (sqrt(3)/2) cos(x) (1/2) = (sin(x) sqrt(3) cos(x)) / 2 = (sin(x) sqrt(3) cos(x)) = sin(2x)sqrt(3) = (2sqrt(3)/2) = sqrt(3) = This equation is false, so the initial statement is not true.
б) sin(x/2) sin(3x/2) = 1/(sin(x)cos(x)/2) (3sin(x)cos(x)/2) = 1/(3sin^2(x)cos^2(x))/4 = 1/(3sin^2(x)(1-sin^2(x)))/4 = 1/(3sin^2(x) - 3sin^4(x))/4 = 1/3sin^2(x) - 3sin^4(x) = 3sin^2(x) - 3(1 - cos^2(x))^2 = (23sin^2(x) - 3(1 - cos(x))^2 = (26sin^2(x) - 3 = 6sin^2(x) = sin^2(x) = 5/sin(x) = sqrt(5/6This equation is correct.
в) 2sin(π/4 + x) sin(π/4 - x) + sin^2(x) = 2(sin(π/4)cos(x) + cos(π/4)sin(x)) (sin(π/4)cos(x) - cos(π/4)sin(x)) + sin^2(x) = 2((sqrt(2)/2)cos(x) + (sqrt(2)/2)sin(x)) ((sqrt(2)/2)cos(x) - (sqrt(2)/2)sin(x)) + sin^2(x) = 2(sqrt(2)(cos^2(x) - sin^2(x))/2) + sin^2(x) = 2(sqrt(2)(1 - 2sin^2(x))) + sin^2(x) = 2sqrt(2) - 4sqrt(2)sin^2(x) + sin^2(x) = (2sqrt(2) - 3sin^2(x)) = 3sin^2(x) = 2sqrt(2sin^2(x) = 2sqrt(2)/sin(x) = sqrt(2/3 sqrt(2)This equation is correct.
а) sin(x-π/3) cos(x-π/6) =
sin(x)cos(π/3)cos(x)sin(π/6) =
sin(x) (sqrt(3)/2) cos(x) (1/2) =
(sin(x) sqrt(3) cos(x)) / 2 =
(sin(x) sqrt(3) cos(x)) =
sin(2x)sqrt(3) =
(2sqrt(3)/2) =
sqrt(3) =
This equation is false, so the initial statement is not true.
б) sin(x/2) sin(3x/2) = 1/
(sin(x)cos(x)/2) (3sin(x)cos(x)/2) = 1/
(3sin^2(x)cos^2(x))/4 = 1/
(3sin^2(x)(1-sin^2(x)))/4 = 1/
(3sin^2(x) - 3sin^4(x))/4 = 1/
3sin^2(x) - 3sin^4(x) =
3sin^2(x) - 3(1 - cos^2(x))^2 = (2
3sin^2(x) - 3(1 - cos(x))^2 = (2
6sin^2(x) - 3 =
6sin^2(x) =
sin^2(x) = 5/
sin(x) = sqrt(5/6
This equation is correct.
в) 2sin(π/4 + x) sin(π/4 - x) + sin^2(x) =
2(sin(π/4)cos(x) + cos(π/4)sin(x)) (sin(π/4)cos(x) - cos(π/4)sin(x)) + sin^2(x) =
2((sqrt(2)/2)cos(x) + (sqrt(2)/2)sin(x)) ((sqrt(2)/2)cos(x) - (sqrt(2)/2)sin(x)) + sin^2(x) =
2(sqrt(2)(cos^2(x) - sin^2(x))/2) + sin^2(x) =
2(sqrt(2)(1 - 2sin^2(x))) + sin^2(x) =
2sqrt(2) - 4sqrt(2)sin^2(x) + sin^2(x) =
(2sqrt(2) - 3sin^2(x)) =
3sin^2(x) = 2sqrt(2
sin^2(x) = 2sqrt(2)/
sin(x) = sqrt(2/3 sqrt(2)
This equation is correct.