Why Do Commutators Have Factors of 1/2?

[Bravus]

Homework Statement

Here's a link to an image of the exam question. It appears in the exam every couple of years, and it's due in my exam this coming week. I've looked in both the textbook and the course notes, and they simply *state* the conclusion, so I don't have a way of proving it, and in particular don't know where the factors of 1/2 come from.

http://www.bravus.com/commutators1.jpg

Homework Equations

See above.

The Attempt at a Solution

Here's an image of my attempts so far:

http://www.bravus.com/commutator2.jpg

 

[vela]

You have $[X,Y]+\{X,Y\}=(XY-YX)+(XY+YX)=2XY$, right? See where the 1/2 comes from now?

 

[Bravus]

Yep, that works, thanks! (As ever, it looks easy once someone explains!)