Compute the number of degrees in the smallest positive angle $x$ such that
\[8 \sin x \cos^5 x - 8 \sin^5 x \cos x = 1.\]
Using the double angle formula, we can write
\begin{align*}
8 \sin x \cos^5 x - 8 \sin^5 x \cos x &= 8 \sin x \cos x (\cos^4 x - \sin^4 x) \\
&= 8 \sin x \cos x (\cos^2 x + \sin^2 x)(\cos^2 x - \sin^2 x) \\
&= 4 \sin 2x \cos 2x \\
&= 2 \sin 4x,
\end{align*}so $\sin 4x = \frac{1}{2}.$  Since $\sin 30^\circ = \frac{1}{2},$ the smallest such $x$ is $\boxed{7.5^\circ}.$