cos 2x = sin x + cos x
Cos^2 x - sin^2 x = cos x + sin x
(cosx + sinx)(cosx - sinx) = cosx + sin x
cancel((cosx + sinx))(cosx - sinx) =cancel( (cosx + sin x))
cos x - sin x = 1
Square both sides
(cos x - sin x)^2 = 1
cos ^2 x + sin ^2 x -2sinx cosx = 1
cancel1 - 2sinx cos x = cancel1
sin 2x = 0
=> 2x = 0=> x = 0
Another method
cos 2x - sin x - cos x = 0
cos^2x - sin^2x -(sinx + cosx) = 0
(Cosx - sinx)(cosx + sinx) -(sinx + cosx) = 0
color(blue)((sinx + cosx))color(red)((cosx - sinx -1)) = 0
color(blue)("Blue part")
color(blue)((sinx + cosx)) = 0
color(blue)(sinx = - cos x)
tanx = -1
x = 135,315
color(red)("red part")
color(red)((cosx - sinx -1)) = 0
color(red)(cos x - sin x = 1
Square both sides
(cos x - sin x)^2 = 1
cos ^2 x + sin ^2 x -2sinx cosx = 1
cancel1 - 2sinx cos x = cancel1
sin 2x = 0
=> 2x = 0,180,360=> x = 0,90,180