To solve the given equation, let's first substitute ctg 2x as y:

y = ctg 2x

Now, the equation becomes:

y² - 6y + 5 = 0

This is a quadratic equation that can be factored as:

(y - 5)(y - 1) = 0

Setting each factor to zero gives us the solutions for y:

y = 5 or y = 1

Now, substitute back y = ctg 2x:

ctg 2x = 5 or ctg 2x = 1

To solve for x, we need to find the angles whose cotangent values are 5 and 1. The values are not standard angles, so we'll use the inverse cotangent function to find the solutions:

For ctg 2x = 5:
2x = arccot(5)
x = (1/2)arccot(5)

For ctg 2x = 1:
2x = arccot(1)
2x = π/4
x = π/8

Therefore, the solutions to the equation ctg²2x - 6ctg 2x + 5 = 0 are x = (1/2)arccot(5) and x = π/8.