Does the graph of $\cos x$ intersect the unit circle other than the point (0,1)? It would seem the unit circle is nicely tucked under the graph of $\cos x$, touching only at (0,1), but is that what's truly going on here? !enter image description here

The slightly brute force approach works quickly. There is symmetry across the $y$-axis, so we can assume that $x\ge 0$. We want to show that for $0 \sqrt{1-x^2}$, or equivalently $\cos^2 x > 1-x^2$ or equivalently $1-\cos^2 x < x^2$ or equivalently $\sin^2 x < x^2$. But this last inequality is probably familiar, perhaps in the form $0\le \frac{\sin x}{x}< 1$ if $x\
e 0$.