1. The range of the function $y=|\cos x|-\cos 2 x(x \in \mathbf{R})$ is
Answer $\left[0, \frac{9}{8}\right]$.
Analysis
$$
y=|\cos x|-\cos 2 x=-2|\cos x|^{2}+|\cos x|+1=-2\left(|\cos x|-\frac{1}{4}\right)^{2}+\frac{9}{8}
$$

Since $0 \leq|\cos x| \leq 1$, we have $0 \leq y \leq \frac{9}{8}$.