We consider wave equations in three space dimensions, and obtain new weighted
$L^\infty$-$L^\infty$ estimates for a tangential derivative to the light cone.
As an application, we give a new proof of the global existence theorem, which
was originally proved by Klainerman and Christodoulou, for systems of nonlinear
wave equations under the null condition. Our new proof has the advantage of
using neither the scaling nor the pseudo-rotation operators.