Evaluate $\tan 10^\circ \tan 20^\circ \tan 30^\circ \tan 40^\circ \tan 50^\circ \tan 60^\circ \tan 70^\circ \tan 80^\circ.$
I recall that $\tan (90^\circ - \theta) = \frac{1}{\tan \theta}$ for any acute angle $\theta$. This means that $\tan 10^\circ = \frac{1}{\tan 80^\circ}$, $\tan 20^\circ = \frac{1}{\tan 70^\circ}$, and so on. So I can pair up the factors in the product and simplify them. For example, $\tan 10^\circ \tan 80^\circ = \frac{1}{\tan 80^\circ} \tan 80^\circ = 1$. Similarly, $\tan 20^\circ \tan 70^\circ = 1$, $\tan 30^\circ \tan 60^\circ = 1$, and $\tan 40^\circ \tan 50^\circ = 1$. So the whole product is just $1 \times 1 \times 1 \times 1 = 1$. 