Answer
$$\frac{{\partial w}}{{\partial x}} = 0$$
Work Step by Step
$$\eqalign{
& {\text{Let }}w = \cos z - \cos x\cos y + \sin x\sin y,\,\,\,\,\,\,\,{\text{ and }}z = x + y \cr
& \cr
& {\text{Write }}w{\text{ in terms of }}x{\text{ and }}y \cr
& w = \cos \left( {x + y} \right) - \cos x\cos y + \sin x\sin y \cr
& w = \cos \left( {x + y} \right) - \left( {\cos x\cos y - \sin x\sin y} \right) \cr
& {\text{Use the trigonometric identity }} \cr
& \cos \left( {\alpha + \beta } \right) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \cr
& {\text{Then,}} \cr
& w = \cos \left( {x + y} \right) - \cos \left( {x + y} \right) \cr
& w = 0 \cr
& {\text{Therefore,}} \cr
& \frac{{\partial w}}{{\partial x}} = 0 \cr} $$
