1) First, let's simplify the equation cos4x + cosx = 0.

Using the trigonometric identity cos(a+b) = cos(a)cos(b) - sin(a)sin(b), we can rewrite the equation as:

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

Let's denote cos(2x) as a, then the equation can be written as:

2(2a^2 - 1)a + cos(x) = 0
4a^3 - 2a + cos(x) = 0

Unfortunately, we can't simplify this further without additional information or using a numerical approach.

2) Next, let's simplify the equation cos5x - cosx = 2sin3x.

Using the trigonometric identity cos(a-b) = cos(a)cos(b) + sin(a)sin(b), we can rewrite the equation as:

2sin(4x)sin(x) + 2sin(3x)cos(3x) = 0
sin(4x + x) + sin(3x + 3x) = 0
sin(5x) + sin(6x) = 0

This equation also cannot be simplified further without additional information or numerical methods.