1. **State the problem:** Simplify the expression $$\frac{x^2 - 2cx + c^2}{x^2 - c^2}$$. 2. **Recognize the formulas:** The numerator is a perfect square trinomial and can be factored as $$(x - c)^2$$. The denominator is a difference of squares and can be factored as $$(x - c)(x + c)$$. 3. **Factor numerator and denominator:** $$\frac{x^2 - 2cx + c^2}{x^2 - c^2} = \frac{(x - c)^2}{(x - c)(x + c)}$$ 4. **Cancel common factors:** $$\frac{\cancel{(x - c)}(x - c)}{\cancel{(x - c)}(x + c)} = \frac{x - c}{x + c}$$ 5. **Final simplified expression:** $$\frac{x - c}{x + c}$$ **Note:** The simplification is valid for all $x \neq c$ and $x \neq -c$ to avoid division by zero.