To solve this equation, we can first rewrite the terms using trigonometric identities. We know that sin(2x) = 2sin(x)cos(x) and sin(8x) = 2sin(4x)cos(4x). We also know that cos(3x) = cos(2x+x) = cos(2x)cos(x) - sin(2x)sin(x).

Now let's rewrite the equation:

2sin(x)cos(x) + 2sin(4x)cos(4x) = √2(cos(2x)cos(x) - sin(2x)sin(x))

Expanding the right side:

2sin(x)cos(x) + 2sin(4x)cos(4x) = √2(cos(2x)cos(x)) - √2(sin(2x)sin(x))

Now, we will expand and simplify this equation further. Since sin(4x) = 2sin(2x)cos(2x), we can rewrite sin(4x)cos(4x) as 2sin(2x)cos^2(2x). We can also decompose cos(2x)cos(x) as [cos(2x + x) + cos(2x - x)]/2 = [cos(3x) + cos(x)]/2.

After making these substitutions, we will have an equation that can be solved using trigonometric identities.