We have to find x for which cos 4x = cos 2x

we use the relation cos 2x = 2(cos x)^2 - 1

cos 4x = cos 2x

=> 2(cos 2x)^2 - 1= cos 2x

=> 2(cos 2x)^2 - cos 2x - 1 = 0

=> 2(cos 2x)^2 - 2cos 2x + cos 2x - 1 = 0

=> 2cos 2x( cos 2x - 1) + 1( cos 2x - 1) = 0

=> ( 2cos 2x + 1)( cos 2x - 1) = 0

( 2cos 2x + 1) = 0

=> cos 2x = -1/2

=> 2x = 120 degrees

=> x = 60 degrees

cos 2x - 1 = 0

=> cos 2x = 1

=> x = 0

**Therefore x = 0 + n*360 degrees and 60+ n*360 degrees**