1. **State the problem:** Simplify the expression $$\left(x-\frac{1}{x}\right)^2$$. 2. **Formula used:** The square of a binomial $$ (a-b)^2 = a^2 - 2ab + b^2 $$. 3. **Apply the formula:** Let $$a = x$$ and $$b = \frac{1}{x}$$. $$\left(x-\frac{1}{x}\right)^2 = x^2 - 2 \cdot x \cdot \frac{1}{x} + \left(\frac{1}{x}\right)^2$$ 4. **Simplify each term:** - $$x^2$$ remains as is. - $$-2 \cdot x \cdot \frac{1}{x} = -2$$ because $$x$$ and $$\frac{1}{x}$$ cancel out. - $$\left(\frac{1}{x}\right)^2 = \frac{1}{x^2}$$. 5. **Combine all terms:** $$x^2 - 2 + \frac{1}{x^2}$$ 6. **Final answer:** $$\boxed{x^2 - 2 + \frac{1}{x^2}}$$