1. **State the problem:** Solve the equation $\tan x + 1 = 0$ using the Regula Falsi method. 2. **Rewrite the equation:** We want to find $x$ such that $f(x) = \tan x + 1 = 0$. 3. **Choose initial interval:** The tangent function has vertical asymptotes at $x = \frac{\pi}{2} + k\pi$, so choose an interval where $f(x)$ changes sign. For example, between $x = -\frac{3\pi}{4}$ and $x = -\frac{\pi}{4}$. 4. **Evaluate function at endpoints:** - $f\left(-\frac{3\pi}{4}\right) = \tan\left(-\frac{3\pi}{4}\right) + 1 = 1 + 1 = 2$ - $f\left(-\frac{\pi}{4}\right) = \tan\left(-\frac{\pi}{4}\right) + 1 = -1 + 1 = 0$ Since $f\left(-\frac{\pi}{4}\right) = 0$, we have found the root exactly at $x = -\frac{\pi}{4}$. 5. **If exact root was not found, apply Regula Falsi formula:** $$x_{new} = b - \frac{f(b)(a - b)}{f(a) - f(b)}$$ where $a$ and $b$ are interval endpoints with $f(a)f(b) < 0$. 6. **Explanation:** Regula Falsi method uses a secant line between points $(a, f(a))$ and $(b, f(b))$ to approximate the root. 7. **Final answer:** The root of $\tan x + 1 = 0$ in the chosen interval is $$x = -\frac{\pi}{4}.$$