To simplify the given expression, we can start by evaluating the trigonometric functions at π/6 and π/4:

sin(π/6) = 1/2
cos(π/6) = √3/2
ctg(π/4) = 1/tan(π/4) = 1
tan(π/4) = 1

Now substitute the values back into the expression:

2sin(π/6) + 5ctg^2 (π/4) + tg(π/4) + 6cos(π/6)
= 2(1/2) + 5(1)^2 + 1 + 6(√3/2)
= 1 + 5 + 1 + 3√3
= 7 + 3√3

Therefore, the simplified expression is 7 + 3√3.