#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#