Sum, Difference, and Multiple-Angle Formulas · Superchallenge problem

The sum $\tan 1^\circ\tan 2^\circ + \tan 2^\circ\tan 3^\circ + \tan 3^\circ\tan 4^\circ + \cdots + \tan 88^\circ\tan 89^\circ$ (with $88$ terms) equals $\cot^2 1^\circ - N$ for an integer $N$. Find $N$.