Jones-Wenzl partial trace ($\mathrm{ptr}(f_n)=({\Delta }_n/{\Delta }_{n-1})f_{n-1}$)

Close the last strand of the projector recursion. Closing $f_{n-1}\otimes 1$ gives $\delta f_{n-1}$, while closing its cap-cup correction gives $f_{n-1}$. The coefficient is $\delta -{\Delta }_{n-2}/{\Delta }_{n-1}={\Delta }_n/{\Delta }_{n-1}$. Iteration proves $\mathrm{tr}(f_n)={\Delta }_n$ and supplies the normalization needed to split the two fusion of Jones-Wenzl colors channels in $f_{n-1}\otimes 1$.