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, 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.
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.