To solve the equation, we can use the double-angle formula for cosine:

cos(2θ) = 2cos^2(θ) - 1

Let's denote x as θ to make it easier. So, the equation becomes:

cos(0.5x) + cos(0.25x) = 0

Using the double-angle formula for cosine, we get:

2cos^2(0.25x) - 1 + 2cos^2(0.125x) - 1 = 0

2cos^2(0.25x) + 2cos^2(0.125x) = 2

cos^2(0.25x) + cos^2(0.125x) = 1

But since cos^2(θ) + sin^2(θ) = 1, we can rewrite the equation as:

sin^2(0.25x) + cos^2(0.125x) = 1

Which means:

sin(0.25x) = 1 and cos(0.125x) = 0

Solving for x:

0.25x = π/2

x = 2π