Trig question pre calc, what am I doing wrong?

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[heavy girder]

First, sec(5π/6) = -2/√(3), not 2/√(3).
Second, (-2√(3)√(2) + 2√(3))/6 = (√(3) - √(6))/3. Not what you did there at the end.

Also, no need to rationalize the denominator everytime.

As you can see, it will still work:
sec(5π/6)cos(7π/4) - cot(-π/3) = -(2/√(3)(1/√(2)) + 1/√(3) = -√(2)/√(3) + 1/√(3) = (1 - √(2))/√(3) = (√(3) - √(6))/3.