1. **Problem:** Simplify the expression $ (1 - \cos\theta)(1 + \cos\theta) $. 2. **Formula:** Use the difference of squares identity: $ (a - b)(a + b) = a^2 - b^2 $. 3. **Apply:** Here, $a = 1$ and $b = \cos\theta$, so $$ (1 - \cos\theta)(1 + \cos\theta) = 1^2 - (\cos\theta)^2 = 1 - \cos^2\theta. $$ 4. **Use Pythagorean identity:** $ \sin^2\theta + \cos^2\theta = 1 $, so $$ 1 - \cos^2\theta = \sin^2\theta. $$ 5. **Final answer:** $$ (1 - \cos\theta)(1 + \cos\theta) = \sin^2\theta. $$