1. **Problem:** Simplify the expression $\sqrt{1 + \cos \beta} \times \sqrt{1 - \cos \beta}$. 2. **Formula and rules:** Recall the identity for the difference of squares: $$a^2 - b^2 = (a - b)(a + b)$$. 3. **Intermediate work:** $$\sqrt{1 + \cos \beta} \times \sqrt{1 - \cos \beta} = \sqrt{(1 + \cos \beta)(1 - \cos \beta)}$$ 4. Apply the difference of squares inside the square root: $$= \sqrt{1 - \cos^2 \beta}$$ 5. Use the Pythagorean identity: $$\cos^2 \beta + \sin^2 \beta = 1 \implies 1 - \cos^2 \beta = \sin^2 \beta$$ 6. Substitute: $$= \sqrt{\sin^2 \beta}$$ 7. Simplify the square root: $$= |\sin \beta|$$ **Final answer:** $$\boxed{|\sin \beta|}$$