To solve the equation cos5x + cosx + 2cos3x = 0, we can rewrite the equation using the trigonometric identity cos(5x) = cos(3x + 2x) and cos(3x) = cos(2x + x).

Therefore, the equation becomes:

cos(3x)cos(2x) - sin(3x)sin(2x) + cosx + 2(cos(2x)cosx - sin(2x)sinx) = 0

Expanding this equation further, we get:

cos(3x)cos(2x) - sin(3x)sin(2x) + cosx + 2cos(2x)cosx - 2sin(2x)sinx = 0

Using the trigonometric identities for cosine of sum and difference of angles, we can simplify this equation further to:

cos(5x) + cos(x) = 0

Therefore, the solution to the equation cos5x + cosx + 2cos3x = 0 is cos(5x) + cos(x) = 0.