Докажите тождество

sinx+cosx+√2 = (2√2) cos² (x/2-П/8)

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