A trigonometric sum

Evaluate

$\displaystyle \mathcal{S} = \frac{1}{\cos 0^\circ \cos 1^\circ}+\frac{1}{\cos 1^\circ \cos 2^\circ}+\cdots+ \frac{1}{\cos 88^\circ \cos 89^\circ}$

Solution

We note that

$\begin{aligned} \frac{1}{\cos k \cos (k+1)} &=\frac{1}{\sin 1}\frac{\sin 1}{\cos k \cos (k+1)} \\ &= \frac{1}{\sin 1}\frac{\sin (k+1)\cos k - \sin k \cos (k+1)}{\cos k \cos (k+1)}\\ &= \frac{1}{\sin 1}\left [ \tan (k+1)- \tan k \right ] \end{aligned}$