Solution:

Multiply the sum by $2 \sin 2.5^\circ$ and apply $2 \sin 5k^\circ \sin 2.5^\circ = \cos(5k - 2.5)^\circ - \cos(5k + 2.5)^\circ,$ so the sum telescopes: $$2 \sin 2.5^\circ \sum_{k=1}^{35} \sin 5k^\circ = \cos 2.5^\circ - \cos 177.5^\circ = 2\cos 2.5^\circ.$$

Hence the sum equals $\frac{\cos 2.5^\circ}{\sin 2.5^\circ} = \cot 2.5^\circ = \tan 87.5^\circ = \tan \frac{175}{2}^\circ.$ Since $\gcd(175, 2) = 1$ and $\frac{175}{2} \lt 90,$ we get $m + n = 175 + 2 = 177.$