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

Therefore, the expression simplifies to:

-sqrt(2)/2 + 3(1/2) - sqrt(3)/3 + sqrt(3)
= -sqrt(2)/2 + 3/2 - sqrt(3)/3 + sqrt(3)
= -sqrt(2)/2 + 3/2 - sqrt(3)/3 + sqrt(3)
= (3-sqrt(2) + 3sqrt(3) - sqrt(3))/2
= (3 + 3sqrt(3) - sqrt(2) - sqrt(3))/2

So, the simplified expression is (3 + 3*sqrt(3) - sqrt(2) - sqrt(3))/2.