1. **State the problem:** Evaluate $\tan\left(\frac{\pi}{4} + \pi\right)$. 2. **Recall the tangent addition formula:** $$\tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}$$ 3. **Apply the formula:** Here, $a = \frac{\pi}{4}$ and $b = \pi$. Since $\tan \pi = 0$, the formula simplifies to: $$\tan\left(\frac{\pi}{4} + \pi\right) = \frac{\tan \frac{\pi}{4} + 0}{1 - \tan \frac{\pi}{4} \times 0} = \tan \frac{\pi}{4}$$ 4. **Evaluate $\tan \frac{\pi}{4}$:** $$\tan \frac{\pi}{4} = 1$$ 5. **Use periodicity of tangent:** Tangent has period $\pi$, so: $$\tan\left(\frac{\pi}{4} + \pi\right) = \tan \frac{\pi}{4} = 1$$ **Final answer:** $$\boxed{1}$$