What is x if cos4x=cos2x ?

To solve this equation, we'll move all the terms to the
left side:

cos 4x - cos 2x =
0

Since the trigonometric functions are matching , we'll
transform the difference into a product.

2
sin[(4x+2x)/2]*sin [(2x-4x)/2]=0

2 sin3x* sin
(-x)=0

From this product of 2 factors, one or the other
factor is zero.

sin3x=0

This
is an elementary equation:

3x = (-1)^k*arcsin 0 +
k*pi

3x = k*pi

x = k*pi/3,
where k is an integer number.

We'll solve the second
elementary equation:

sin
(-x)=0

-sin (x)=0

sin
x=0

x=(-1)^k arcsin 0 +
k*pi

x=k*pi

The
set of solutions is : S={k*pi/3}U{k*pi}.