Sinx + cosx = 1 - sin 2 х

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