# Exact value of a trigonometric rational

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 $$\cfrac{\sqrt3}{3}$$

• Approaching this problem by substitution would make a little bit more sense if you chose $u=\frac\pi{30}$, so $\tan\frac\pi5=\tan6u$. From here, you can use double and triple angle identities to manipulate $\tan6u$. But as per the hint below, try to recognize that the given expression is the result of a compound angle identity for the tangent ratio. Dec 22, 2019 at 14:59
• If the identity suggested by @YiFan seems unfamiliar, try multiplying your expression by $\frac{\cos \frac{\pi}{5}\cos \frac{\pi}{30}}{\cos \frac{\pi}{5}\cos \frac{\pi}{30}}$. The sum of angle formulas for sine and cosine may be more recognizable. Dec 22, 2019 at 15:54

Hint. Use the tangent angle subtraction formula $$\tan(a-b)=\frac{\tan a-\tan b}{1+\tan a\tan b}.$$