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