1. **State the problem:** Find the exact value of $$\frac{\tan \frac{\pi}{10} + \tan \frac{\pi}{15}}{1 - \tan \frac{\pi}{10} \tan \frac{\pi}{15}}$$. 2. **Recall the formula:** This expression matches the tangent addition formula: $$\tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}$$ where $$a = \frac{\pi}{10}$$ and $$b = \frac{\pi}{15}$$. 3. **Apply the formula:** Using the formula, the expression simplifies to: $$\tan\left(\frac{\pi}{10} + \frac{\pi}{15}\right)$$ 4. **Add the angles:** Find a common denominator for $$\frac{\pi}{10}$$ and $$\frac{\pi}{15}$$: $$\frac{\pi}{10} = \frac{3\pi}{30}, \quad \frac{\pi}{15} = \frac{2\pi}{30}$$ So, $$\frac{3\pi}{30} + \frac{2\pi}{30} = \frac{5\pi}{30} = \frac{\pi}{6}$$ 5. **Evaluate the tangent:** $$\tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}}$$ **Final answer:** $$\boxed{\frac{1}{\sqrt{3}}}$$