Compute $\arcsin(\sin 66^\circ-\sin 54^\circ)$ in degrees.
From the sum-to-product formula,
\[ \sin x- \sin z = 2\sin \frac{x-z}{2}\cos\frac{x+z}{2}.\]Applying this with $x = 66^{\circ}$ and $z = 54^{\circ}$, we have
\begin{align*}
\arcsin(\sin 66^\circ-\sin54^\circ)&=\arcsin \Big(2\sin\frac{66^\circ -54^\circ }{2}\cos\frac{66^\circ +54^\circ }{2} \Big)\\
&=\arcsin(2\sin6^\circ\cos 60^\circ)\\
&=\arcsin(\sin 6^\circ) \\
&= \boxed{6^{\circ}}.
\end{align*}