Explanation:

- To find \( \frac{dy}{dx} \), use the parametric form: \( x = a(\theta - \sin\theta) \) and \( y = a(1 + \cos\theta) \).

- Use the chain rule for parametric equations: \( \frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} \).

- First, calculate \(\frac{dy}{d\theta}\):

- \( y = a(1 + \cos\theta) \)

- \( \frac{dy}{d\theta} = a(-\sin\theta) \)

- Next, calculate \(\frac{dx}{d\theta}\):

- \( x = a(\theta - \sin\theta) \)

- \( \frac{dx}{d\theta} = a(1 - \cos\theta) \)

- Calculate: \( \frac{dy}{dx} = \frac{a(-\sin\theta)}{a(1 - \cos\theta)} = \frac{-\sin\theta}{1 - \cos\theta} \)

- Simplification:

- \( \frac{-\sin\theta}{1 - \cos\theta} = -\cot(\theta/2) \)

- Option 2 is correct: \(- \cot \frac{\theta}{2}\)