Докажите тождество sinx+cosx+√2=(2√2)cos²(x/2-п/8) - вопрос №230227522228 от manilowyn2 24.07.2021 03:56

Question

Алгебра: Докажите тождество sinx+cosx+√2=(2√2)cos²(x/2-п/8)

manilowyn2 · Answer

Cos^2 (Pi/8 - x) - cos^2 (Pi/8 + x) = (cos Pi/8*cos x + sin Pi/8*sin x)^2 - (cos Pi/8*cos x - sin Pi/8*sin x)^2 =
= (cos^2 Pi/8*cos^2 x + 2cos Pi/8*cos x*sin Pi/8*sin x + sin^2 Pi/8*sin^2 x) -
- (cos^2 Pi/8*cos^2 x - 2cos Pi/8*cos x*sin Pi/8*sin x + sin^2 Pi/8*sin^2 x) =
= 4cos Pi/8*cos x*sin Pi/8*sin x = 2sin Pi/8*cos Pi/8 * 2sin x*cos x = sin Pi/4 * sin 2x = V(2)/2 * sin 2x = 1/2
sin 2x = 1/V(2) = V(2)/2
2x1 = Pi/4 + 2Pi*n
2x2 = 3Pi/4 + 2Pi*n

x1 = Pi/8 + Pi*n
x2 = 3Pi/8 + Pi*n