<s>[INST] How to prove that $\left(\sqrt{3}\sec{\frac{\pi}{5}}+\tan{\frac{\pi}{30}}\right)\tan{\frac{2\pi}{15}}=1$ From this geometry problem, I can not find geometry solution.
However the answer is $X=\frac{2\pi}{15}$ by geometry method.

Then I get the identity $$\left(\sqrt{3}\sec{\frac{\pi}{5}}+\tan{\frac{\pi}{30}}\right)\tan{\frac{2\pi}{15}}=1.$$
How to prove it by trigonometric method ?
Thank in advances.
 [/INST] I'd try using that 
$$\frac{1}{5} - \frac{1}{2} + \frac{1}{3}  = \frac{1}{30},$$
$$\frac{4}{3} - \frac{6}{5} = \frac{2}{15}.$$
And the formula:
$$
\tan(a\pm b) = \frac{\tan a \pm \tan b}{1 \mp \tan a\tan b}.
$$
</s>