To solve this expression, we first need to find the exact values of the trigonometric functions at the given angles:

tg(30°) = sin(30°) / cos(30°) = (1/2) / (√3/2) = √3 / 3

cos(30°) = √3/2

sin(60°) = √3/2

cos(45°) = √2/2

tg(60°) = √3

Now substitute these values back into the original expression:

4(√3/3) - 5(√3/2) + 6(√3/2) - 4(√2/2) - √3

Simplify:

(4√3/3) - (5√3/2) + (6√3/2) - (4√2/2) - √3

(8√3/6) - (15√3/6) + (18√3/6) - (12√2/6) - √3

(8√3 - 15√3 + 18√3 - 12√2 - 6√3) / 6

(11√3 - 12√2) / 6

Therefore, the final result is:

(11√3 - 12√2) / 6