A geometry problem by أحمد الحلاق

$\begin{aligned} S & = \log (\tan 3^\circ) + \log(\tan 6^\circ) + \log(\tan 9^\circ) + \cdots + \log( \tan 87^\circ) \\ & = \log (\tan 3^\circ) + \log(\tan 87^\circ) + \log(\tan 6^\circ) + \log( \tan 84^\circ) + \cdots + \log( \tan 42^\circ) + \log( \tan 48^\circ) + \log( \tan 45^\circ) \\ & = \log (\tan 3^\circ) + \log(\cot 3^\circ) + \log(\tan 6^\circ) + \log(\cot 6^\circ) + \cdots + \log( \tan 42^\circ) + \log(\cot 42^\circ) + \log( \tan 45^\circ) \\ & = \log \big(\tan 3^\circ \cdot \cot 3^\circ \cdot \tan 6^\circ \cdot \cot 6^\circ \cdots \tan 42^\circ \cdot \cot 42^\circ \cdot \tan 45^\circ\big) \\ & = \log \big(\tan 3^\circ \cdot \frac 1{\tan 3^\circ} \cdot \tan 6^\circ \cdot \frac 1{\tan 6^\circ} \cdots \tan 42^\circ \cdot \frac 1{\tan 42^\circ} \cdot \tan 45^\circ\big) \\ & = \log \big(\cancel{\tan 3^\circ \cdot \frac 1{\tan 3^\circ}}^1 \cdot \cancel{\tan 6^\circ \cdot \frac 1{\tan 6^\circ}}^1 \cdots \cancel{\tan 42^\circ \cdot \frac 1{\tan 42^\circ}}^1 \cdot \cancel{\tan 45^\circ}^1 \big) \\ & = \log 1 = \boxed{0} \end{aligned}$