Solution:

Multiply numerator and denominator by $2\sin\frac{1}{2}^\circ.$ Since $2\cos n^\circ \sin\frac{1}{2}^\circ = \sin\left(n + \frac{1}{2}\right)^\circ - \sin\left(n - \frac{1}{2}\right)^\circ$ and $2\sin n^\circ \sin\frac{1}{2}^\circ = \cos\left(n - \frac{1}{2}\right)^\circ - \cos\left(n + \frac{1}{2}\right)^\circ,$ both sums telescope: $$x = \frac{\sin 44.5^\circ - \sin 0.5^\circ}{\cos 0.5^\circ - \cos 44.5^\circ} = \frac{2\cos 22.5^\circ \sin 22^\circ}{2\sin 22.5^\circ \sin 22^\circ} = \cot 22.5^\circ,$$ using the sum-to-product identities in the last step.

By the half-angle formula, $\cot 22.5^\circ = \frac{1 + \cos 45^\circ}{\sin 45^\circ} = \sqrt{2} + 1.$ Hence $100x = 100\sqrt{2} + 100 = 241.42\ldots,$ and the greatest integer not exceeding it is $241.$