How would i solve this trig equation?

Question

"cos 2x = cos x if 0 < x <360" its less than or equal to btw. having trouble with a few problems in the math packet : /

Answer 1

where cos x is zero which is at 0 and 180, but these data points are not in the range of the problem, so it must be 120 and 240 degrees

Answer 2

cos 2x = cos x
2 cos^2 x - 1 = cos x
2 cos^2 x - cos x - 1 = 0
let X = cos x, then 2X^2 - X - 1 = 0
solving quadratic equation, X = 1 or X = -1/2
thus, cos x = 1, or x = 0 or 360
cos x = - 1/2 or x = 120 or 240

based on the condition 0 < x < 360, only x = 120 or 240 is qualified

Answer 3

cos 2x = (cos x)^2 - (sin x)^2 = 2(cos x)^2 -1
so,
2(cos x)^2 - 1 = cos x
2(cos x)^2 - cos x - 1 = 0 ---> make cos x = y
2y^2 - y - 1 = 0 ---> use the quadratic formula
y = 1 or y = -0.50
cos x = 1 or cos x = -0.50
x = 0 or 120, but 0 < x < 360, so
x = 120

Answer 4

cos 2x = cos^2x-sin^2x = 2cos^2x-1
so 2cos^2x-cos x-1=0
call cos x= z
and
2z^2 -z-1=0 z=((1+-sqrt(9))/4 so z=1 and z= -1/2
cosx = 1 so x= 0
cos x= -1/2 x= 120º and x= 240º

Answer 5

cos(x)=cos(-x)=cos(360-x)
So
2x=360-x
3x=360
x=120
cos(120)= -1/2
cos(240)= -1/2