Let's break down the expression step by step:

2cos 135° = 2 * cos(135°)

Using the unit circle, we know that the cosine of 135° is -√2/2.

2 * (-√2/2) = -√2

tg (π/3) = tan(π/3)

Using the unit circle, we know that the tangent of π/3 is sqrt(3).

sin (-2π/3) = sin(-2π/3)

Sine function is an odd function, so sin(-x) = -sin(x).

Therefore, sin(-2π/3) = -sin(2π/3) = -sqrt(3)/2

ctg^2 (-2π/3) = cot^2(-2π/3)

Cotangent function is the reciprocal of the tangent function, so cot(x) = 1/tan(x).

Therefore, cot(-2π/3) = 1/tan(2π/3) = 1/sqrt(3) = sqrt(3)/3.

Now, let's substitute these values back into the original expression:

2cos 135° - tg π на 3 sin (-2π на 3) - ctg в квадрате (-2π на 3)
= -√2 - sqrt(3) * (-sqrt(3)/2) - (sqrt(3)/3)^2
= -√2 + 3/2 - 3/9
= -√2 + 3/2 - 1/3
= -√2 + 4/6 - 2/6
= (-√2 + 4 - 2)/6
= (2 - √2)/6

Therefore, the final simplified expression is:
(2 - √2)/6