2) Вычислить tg 2x, если tg x = 1/2
1/cos^2 x = 1 + tg^2 x = 1 + 1/4 = 5/4
cos^2 x = 4/5
cos x = 2/sqrt(5) = 2sqrt(5)/5
sin^2 x = 1 - cos^2 x = 1 - 4/5 = 1/5
sin x = 1/sqrt(5) = sqrt(5)/5
sin 2x = 2sin x*cos x = 2*sqrt(5)/5*2sqrt(5)/5 = 4*5/25 = 4/5
cos 2x = cos^2 x - sin^2 x = 4/5 - 1/5 = 3/5
tg 2x = sin 2x/cos 2x = (4/5) : (3/5) = 4/3
3) sin 2x + (sin x - cos x)^2 =
= 2sin x*cos x + sin^2 x - 2sin x*cos x + cos^2 x =
= sin^2 x + cos^2 x = 1
cos 3x + sin x - sin 2x = cos (2x + x) + sin x - sin 2x =
= cos 2x*cos x - sin 2x*sin x + sin x - sin 2x =
= (cos^2 x - sin^2 x)*cos x - 2sin x*cos x*sin x + sin x - sin 2x =
= (cos^2 x - sin^2 x)*cos x - 2sin^2 x*cos x + sin x - sin 2x =
= cos x*(cos^2 x - 3sin^2 x) + sin x - 2sin x*cos x =
= cos x*(cos^2 x - 3(1 - cos^2 x)) + sin x - 2sin x*cos x =
= cos x*(4cos^2 x - 3) + sin x - 2sin x*cos x =
= 4cos^3 x - 3cos x + sin x - 2sin x*cos x
Больше это сократить никак не получается.
Может быть, в задании ошибка?