10. Let $x+\sin x \cdot \cos x-1=0$, $2 \cos y-2 y+\pi+4=0$. Then the value of $\sin (2 x-y)$ is
10. -1 .

From $x+\sin x \cdot \cos x-1=0$, we get
$$
2 x+\sin 2 x=2 \text {; }
$$

From $2 \cos y-2 y+\pi+4=0$, we get
$$
\frac{\pi}{2}-y+\sin \left(\frac{\pi}{2}-y\right)=-2 \text {. }
$$

Let $f(x)=x+\sin x$. Then
$$
f^{\prime}(x)=1-\cos x \geqslant 0 \text {. }
$$

Thus, $f(x)$ is monotonically increasing on $\mathbf{R}$ and is an odd function.
From equations (1) and (2), we get
$$
\begin{array}{l}
f(2 x)=2, f\left(\frac{\pi}{2}-y\right)=-2 \\
\Rightarrow 2 x+\frac{\pi}{2}-y=0 \Rightarrow 2 x-y=-\frac{\pi}{2} \\
\Rightarrow \sin (2 x-y)=-1
\end{array}
$$