Question

Sinx + cosx= 1- sin 2 х

Answer 1

Sin x + cos x = cos^2 x + sin^2 x - 2sin x*cos x
sin x + cos x = (sin x - cos x)^2
Можно написать так
sin x + cos x = √2*(sin x*1/√2 + cos x*1/√2) =
= √2*(sin x*cos pi/4 + cos x*sin pi/4) = √2*sin (x + pi/4)
Точно так же
sin x - cos x = √2*sin (x - pi/4)
Подставляем
√2*sin (x + pi/4) = (√2*sin(x - pi/4))^2 = 2sin^2 (x - pi/4)
Но sin (x + pi/4) = sin (x + pi/2 - pi/4) = cos (x - pi/4)
√2*cos (x - pi/4) = 2 - 2cos^2 (x - pi/4)
Замена cos (x - pi/4) = y
√2*y = 2 - 2y^2
2y^2 + √2*y - 2 = 0
D = (√2)^2 - 4*2(-2) = 2 + 16 = 18 = (3√2)^2
y1 = cos (x - pi/4) = (-√2 - 3√2)/4 = - 4√2/4 = -√2 < -1 - не подходит
y2 = cos (x - pi/4) = (-√2 + 3√2)/4 = √2/2
x1 - pi/4 = pi/4 + 2pi*k; x1 = pi/2 + 2pi*k
x2 - pi/4 = -pi/4 + 2pi*k; x2 = 2pi*k