Answer
$$W=-2 \mathrm{e}^{x} x\cos^{2} x -2\mathrm{e}^{x} x \sin^{2} x +2 \mathrm{e}^{x}\cos^{2} x+2 \mathrm{e}^{x}\sin^{2} x.$$
Work Step by Step
The Wronskian of $\left\{x, e^{x}, \sin x, \cos x\right\}$ is given by
$$W=\left|\begin{array}{ccc}{x}&{ \mathrm{e}^{x}} & {\sin x} & {\cos x} \\ {1} & {\mathrm{e}^{x}} & {\cos x} & {-\sin x} \\ {0} & {\mathrm{e}^{x}} & {-\sin x} & {-\cos x} \\ {0} & {\mathrm{e}^{x}} & {-\cos x} & {\sin x}\end{array}\right|\\
=-2 \mathrm{e}^{x} x\cos^{2} x -2\mathrm{e}^{x} x \sin^{2} x +2 \mathrm{e}^{x}\cos^{2} x+2 \mathrm{e}^{x}\sin^{2} x .$$
