Intuition/Logic for Sum of Arctangent Problem

Hi all -

I realize this problem might just look like something from precalculus --- but because I'm not asking about the actual solution but rather the "analysis" of one part of it --- I thought to post here. I'm pretty comfortable with my own solution.

But my own work oesn't actually tell/explain/show me why the hint actually works. Can someone please explain how I could come up/infer/guess/calculate the hint by myself --- if the question didn't give it to me? All I can understand is that the problem is trying to use the result in the second expression to prove the last expression. Thank you---

My work which looks right:

**Actual question ---**

http://i47.tinypic.com/28i7wix.png