Question

Докажите тождество
sinx+cosx+√2=(2√2)cos²(x/2-П/8)

Answer 1

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