In any triangle ABC, the expression b·cosC + c·cosB simplifies to:

Drop the altitude from A onto BC: it splits side a into the two pieces $c$·cosB and $b$·cosC, so their sum rebuilds a. (This is the projection formula.) Check on a 3-4-5 triangle with $a=3,b=4,c=5$ and $C=90$°: $4\times 0+5\times$ cosB = $5\times 0.6=3=a$. Answering $b+c$ would require both cosines to be 1, i.e. a degenerate triangle.