1. **Problem:** Simplify the function $$f(x) = \frac{\sin x}{1 - \cos x}$$ for $$x \neq 2\pi k, k \in \mathbb{Z}$$. 2. **Formula and identities used:** - Use the half-angle identity: $$1 - \cos x = 2 \sin^2 \left( \frac{x}{2} \right)$$. - Use the double-angle identity for sine: $$\sin x = 2 \sin \left( \frac{x}{2} \right) \cos \left( \frac{x}{2} \right)$$. 3. **Step-by-step simplification:** - Substitute the identities into the function: $$f(x) = \frac{2 \sin \left( \frac{x}{2} \right) \cos \left( \frac{x}{2} \right)}{2 \sin^2 \left( \frac{x}{2} \right)}$$ - Cancel the common factor 2: $$f(x) = \frac{\sin \left( \frac{x}{2} \right) \cos \left( \frac{x}{2} \right)}{\sin^2 \left( \frac{x}{2} \right)}$$ - Simplify the fraction: $$f(x) = \frac{\cos \left( \frac{x}{2} \right)}{\sin \left( \frac{x}{2} \right)} = \cot \left( \frac{x}{2} \right)$$ 4. **Explanation:** - We used trigonometric identities to rewrite the numerator and denominator in terms of half angles. - After substitution, common factors were canceled, leading to the cotangent of half the angle. **Final answer:** $$f(x) = \cot \left( \frac{x}{2} \right)$$