What are x values for cos 2x = cos x?

We'll subtract cos x both sides and we'll
get:

cos 2x - cos x = 0

Now,
we'll apply the formula for the double angle 2x:

cos 2x =
cos (x+x) = cos x*cos x - sin x*sin x

cos 2x = (cos x)^2 -
(sin x)^2

We'll write (sin x)^2 = 1 - (cos x)^2
(fundamental formula of trigonometry).

cos 2x = (cos x)^2
- [1 - (cos x)^2] 

We'll remove the
brackets:

cos 2x = (cos x)^2 - 1 + (cos
x)^2

We'll combine like
terms:

cos 2x = 2(cos x)^2 - 1
(1)

We'll re-write the equation, substituting cos 2x y the
expression (1).

2(cos x)^2 - 1 - cos x =
0

Now , we'll use substitution technique to solve the
equation.

We'll note cos x = t and we'll re-write the
equation in t:

2t^2 - t - 1 =
0

Since it is a quadratic, we'll apply the quadratic
formula:

t1 = {-(-1) + sqrt[(-1)^2 +
4*2*1]}/2*2

t1 =
[1+sqrt(1+8)]/4

t1 =
(3+1)/4

t1 = 1

t2 =
(1-3)/4

t2 = -1/2

Now, we'll
put cos x = t1.

cos x =
1

Since it is an elementary equation, we'll apply the
formula:

cos x = a

x = +/-
arccos a + 2k*pi

x = arccos 1 +
2k*pi

x = 0

x =
2pi

cos x = t2

cos x =
-1/2

x = pi - pi/3

x =
2pi/3

x = pi + pi/3

x =
4pi/3

The solutions for the equation are:{0 ;
2pi/3 ; 4pi/3 ; 2pi}.