Answer
$$W= \mathrm{e}^{-x} \cos x-\mathrm{e}^{-x} \sin x.$$
Work Step by Step
The Wronskian of $\left\{1, x, \cos x, e^{-x}\right\}$ is given by
$$W=\left|\begin{array}{cccc}{1} & {x} & {\cos x} & {\mathrm{e}^{-x}} \\ {0} & {1} & {-\sin x} & {-\mathrm{e}^{-x}} \\ {0} & {0} & {-\cos x} & {\mathrm{e}^{-x}} \\ {0} & {0} & {\sin x} & {-\mathrm{e}^{-x}}\end{array}\right|= \mathrm{e}^{-x} \cos x-\mathrm{e}^{-x} \sin x.$$
