Answer
$$
z_x= \sin( x) \sinh(t-\cos x) ,\quad
z_t= \sinh(t-\cos x) .
$$
Work Step by Step
Recall that $(\cosh x)'=\sinh x$
Recall that $(\cos x)'=-\sin x$.
Since $ z=\cosh (t-\cos x)$, by using the chain rule, we have
$$
z_x= \sinh(t-\cos x) (\sin x)=\sin( x) \sinh(t-\cos x) ,\\
z_t= \sinh(t-\cos x) (1)= \sinh(t-\cos x) .
$$
