Finding a composite inverse function.

Hey guys! I need to find:

Here's how I'm going about finding the answer:

I read this as, "Tangent of the angle whose cotangent is negative 1 over the square root of 3."

As I understand it, I need to draw the triangle for the inverse cotangent in quadrant 4 since it is negative. Here's my drawing:

http://i43.tinypic.com/s2e5bc.jpg

Now, using this drawing, I would find the tangent (and the answer to the expression) to be:

Did I do this correctly? I'm having a hard time with the concept of inverse functions in general, so any advice or somewhere I could read to better understand the concept would be helpful. I've read my trig book and it's not helping me very much.

I'm doubting that it is correct because when I evaluate the expression on my TI-84, the answer doesn't match.

Thanks for any help!