Trigonometric pattern

We note that $6! = 720$ . This means that $n!$ is a multiple of 180 for $n \ge 0$ and $tan n!^\circ = 0$ for $n \ge 6$ . Therefore,

$\begin{aligned} X & = \sum_{n=1}^{2018} \tan n!^\circ - \prod_{n=1}^{2018} \tan n!^\circ \\ & = \sum_{n=1}^{\color{#D61F06}5} \tan n!^\circ - 0 \\ & = \tan 1^\circ + \tan 2^\circ + \tan 6^\circ + \tan 24^\circ + \tan 120^\circ \\ & \approx - 1.129 \end{aligned}$

Therefore, $\lfloor X \rfloor = \boxed{-2}$ .