1. **State the problem:** Simplify the expression $$\frac{1-\cos x}{\sin x} \times \frac{1}{\cos x}$$. 2. **Recall the formula and rules:** We will use trigonometric identities and algebraic simplification. Important identities: - $$\sin^2 x + \cos^2 x = 1$$ - We can factor and cancel common terms carefully. 3. **Rewrite the expression:** $$\frac{1-\cos x}{\sin x} \times \frac{1}{\cos x} = \frac{1-\cos x}{\sin x \cos x}$$ 4. **Use the identity for the numerator:** Note that $$1-\cos x = 2\sin^2 \frac{x}{2}$$. So, $$\frac{1-\cos x}{\sin x \cos x} = \frac{2\sin^2 \frac{x}{2}}{\sin x \cos x}$$ 5. **Express $$\sin x$$ in terms of half-angle:** $$\sin x = 2 \sin \frac{x}{2} \cos \frac{x}{2}$$. 6. **Substitute $$\sin x$$:** $$\frac{2\sin^2 \frac{x}{2}}{(2 \sin \frac{x}{2} \cos \frac{x}{2}) \cos x} = \frac{2\sin^2 \frac{x}{2}}{2 \sin \frac{x}{2} \cos \frac{x}{2} \cos x}$$ 7. **Cancel common factors:** $$= \frac{\cancel{2} \sin^2 \frac{x}{2}}{\cancel{2} \sin \frac{x}{2} \cos \frac{x}{2} \cos x} = \frac{\sin \frac{x}{2}}{\cos \frac{x}{2} \cos x}$$ 8. **Final simplified form:** $$\boxed{\frac{\sin \frac{x}{2}}{\cos \frac{x}{2} \cos x}}$$ This is the simplified expression in terms of half-angle functions.