Chứng minh mỗi đẳng thức sau là đúng:

$\tan \frac{{9\pi }}{{20}} = \frac{{1 + \tan \frac{\pi }{5}}}{{1 - \tan \frac{\pi }{5}}}$.

Ta có$\tan \frac{{9\pi }}{{20}} = \tan \left( {\frac{\pi }{4} + \frac{\pi }{5}} \right) = \frac{{\tan \frac{\pi }{4} + \tan \frac{\pi }{5}}}{{1 - \tan \frac{\pi }{4}.\tan \frac{\pi }{5}}} = \frac{{1 + \tan \frac{\pi }{5}}}{{1 - \tan \frac{\pi }{5}}}$.

Vậy $\tan \frac{{9\pi }}{{20}} = \frac{{1 + \tan \frac{\pi }{5}}}{{1 - \tan \frac{\pi }{5}}}$ (đpcm).