1. **State the problem:** Solve the equation $$\tan^2 x - \tan x = 0$$ for $$-\pi < x < \pi$$. 2. **Rewrite the equation:** Factor the left side: $$\tan x (\tan x - 1) = 0$$ 3. **Set each factor equal to zero:** - $$\tan x = 0$$ - $$\tan x - 1 = 0 \implies \tan x = 1$$ 4. **Solve $$\tan x = 0$$:** The tangent function is zero at integer multiples of $$\pi$$: $$x = k\pi$$ for integer $$k$$. Within $$-\pi < x < \pi$$, the solutions are: $$x = 0$$ 5. **Solve $$\tan x = 1$$:** Tangent equals 1 at angles where: $$x = \frac{\pi}{4} + k\pi$$ for integer $$k$$. Within $$-\pi < x < \pi$$, the solutions are: $$x = -\frac{3\pi}{4}, \frac{\pi}{4}$$ 6. **Combine all solutions:** $$x = -\frac{3\pi}{4}, 0, \frac{\pi}{4}$$ **Final answer:** $$x = -\frac{3\pi}{4}, 0, \frac{\pi}{4}$$