I present a method of calculating the coefficients appearing in the Jones-Wenzl projections in the Temperley-Lieb algebras. It essentially repeats the approach of Frenkel and Khovanov in  published in 1997. I wrote this note mid-2002, not knowing about their work, but then set it aside upon discovering their article.
Recently I decided to dust it off and place it on the arXiv — hoping the self-contained and detailed proof I give here may be useful.
The proof is based upon a simplification of the Wenzl recurrence relation. I give an example calculation, and compare this method to the formula announced by Ocneanu  and partially proved by Reznikoff . I also describe certain moves on diagrams which modify their coefficients in a simple way.