I want to find the exact value of $$\cfrac{\tan \cfrac{\pi}{5}-\tan\cfrac{\pi}{30}}{1+\tan \cfrac{\pi}{5}\cdot \tan \cfrac{\pi}{30}}$$
I started with $u$ substitution, where $u=\pi/5$, and therefore $\cfrac u6 =\pi/30$, allowing me to rewrite the problem as
\begin{align*} \cfrac{\tan u – \tan \cfrac u6}{1+\tan u \cdot \tan \cfrac u6}
\end{align*}
I tried dividing both sides of the rational by trig functions like $\tan u$ or $\cos u/6$, new definitions (e.g., trig Pythagorean identity for 1), and I tried using a calculator and I would only get decimal values when I'm trying to get the answer $\cfrac{\sqrt3}{3}$
Best Answer
Hint. Use the tangent angle subtraction formula $$\tan(a-b)=\frac{\tan a-\tan b}{1+\tan a\tan b}.$$