To solve the trigonometric equation cos4x + cos8x + cos12x = 0, we can make use of the sum-to-product trigonometric identities.

Using the identity cos(A) + cos(B) = 2cos((A + B)/2)cos((A - B)/2), we can rewrite the equation as:

2cos(6x)cos(-2x) + cos(6x) = 0

Simplifying this further, we get:

cos(6x)(2cos(-2x) + 1) = 0
cos(6x)(2cos(2x) + 1) = 0

The solutions can be found by setting each factor to zero:

1) cos(6x) = 0
6x = π/2 + nπ, where n is an integer
x = π/12 + (n/6)π

2) 2cos(2x) + 1 = 0
cos(2x) = -1/2
2x = 2π/3 + 2nπ or 2x = 4π/3 + 2nπ

x = π/3 + nπ or x = 2π/3 + nπ

Therefore, the solutions to the equation cos4x + cos8x + cos12x = 0 are:

x = π/12 + (n/6)π, π/3 + nπ, 2π/3 + nπ
where n is an integer.