To simplify the given equation:

cos 2x - 3 cos x + 1 = 1 /[(ctg 2x - ctg x) sin (x - π)]

First, let's simplify the left side of the equation:

cos 2x - 3 cos x + 1

Using the double angle formula for cosine (cos 2θ = 2 cos^2 θ - 1), we can rewrite cos 2x as:

2 cos^2 x - 1 - 3 cos x + 1
= 2 cos^2 x - 3 cos x

Now, let's simplify the right side of the equation:

1 /[(ctg 2x - ctg x) sin (x - π)]

Using trigonometric identities, we know that cotangent is the reciprocal of tangent:

ctg 2x = 1 / tan 2x
ctg x = 1 / tan x

So, we can rewrite the right side of the equation as:

1 /[(1/tan 2x - 1/tan x) sin (x - π)]
= 1 /[(tan x - tan x) sin (x - π)]
= 1 / [0 sin (x - π)]
= 1 / 0

As sin(x - π) = 0, the expression becomes undefined.

Therefore, the equation cos 2x - 3 cos x + 1 = 1 /[(ctg 2x - ctg x) sin (x - π)] is not valid as it leads to an undefined result.