Find the value of
\[8\cos^210^\circ - \dfrac{1}{\sin 10^\circ}.\]
We can write
\[8 \cos^2 10 ^\circ - \frac{1}{\sin 10^\circ} = \frac{8 \cos^2 10^\circ \sin 10^\circ - 1}{\sin 10^\circ}.\]By the double-angle formula, $2 \cos 10^\circ \sin 10^\circ = \sin 20^\circ,$ so
\[\frac{8 \cos^2 10^\circ \sin 10^\circ - 1}{\sin 10^\circ} = \frac{4 \sin 20^\circ \cos 10^\circ - 1}{\sin 10^\circ}.\]From the product-to-sum formula, $2 \sin 20^\circ \cos 10^\circ = \sin 30^\circ + \sin 10^\circ,$ so
\[\frac{4 \sin 20^\circ \cos 10^\circ - 1}{\sin 10^\circ} = \frac{2 \sin 30^\circ + 2 \sin 10^\circ - 1}{\sin 10^\circ} = \frac{2 \sin 10^\circ}{\sin 10^\circ} = \boxed{2}.\]