A bumpy ride

We cannot roll the ellipse on the cosine curve very far, not even down to the point $(\pi,-3)$ . The "valley" of the cosine graph is too narrow (or, the ellipse is too "wide"). Let's argue indirectly and assume we get that far. When the ellipse touches the cosine graph at $(\pi,-3)$ , its major axis will be horizontal, since we have rolled off half of the circumference of the ellipse (the circumference of the ellipse is equal to the arc length of the cosine graph between 0 and $2\pi$ ). The curvature of the ellipse at $(\pi,-3)$ will be $\frac{1}{2}$ . But the curvature of the cosine graph at that point is 1, a mismatch, showing that this scenario is impossible. Thus the claim is $\boxed{False}$ .