= Solution
Write $z(x,y)=Z(r,s)$ in the new <coordinates>. Since
$$
\partial_x=\partial_r-\partial_s,\qquad \partial_y=\partial_r+\partial_s,
$$
the <partial differential equation> becomes
$$
z_{xx}-z_{yy}=-4Z_{rs}=0.
$$
For a twice continuously differentiable <solution of a differential equation>, $Z_r$ is independent of $s$, so <integration> first in $r$ and then in $s$ gives
$$
\boxed{z(x,y)=\phi(y+x)+\psi(y-x).}
$$
This is the decomposition of the <one-dimensional wave equation> into waves traveling in opposite directions.
The first <initial condition> gives $\phi(y)+\psi(y)=\cos y$. The second <initial condition> gives $\phi'(y)-\psi'(y)=0$. Differentiating the first relation and combining the two relations yields $\phi'=psi'=-\tfrac12\sin y$. Thus $\phi(y)=\tfrac12\cos y+C$ and $\psi(y)=\tfrac12\cos y-C$. The constants cancel, and the <cosine addition formula> gives
$$
\boxed{z(x,y)=\frac{\cos(y+x)+\cos(y-x)}2=\cos x\cos y.}
$$
Both <initial conditions> and the <partial differential equation> follow directly by differentiation.
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