First, find \(\frac{dx}{d\theta}\) and \(\frac{dy}{d\theta}\):

\(\frac{dx}{d\theta} = \\sin \theta\)

\(\frac{dy}{d\theta} = -\\sin \theta + \frac{1}{2} \\sin 2\theta\)

Use the chain rule to find \(\frac{dy}{dx}\):

\(\frac{dy}{dx} = \frac{dy}{d\theta} \div \frac{dx}{d\theta} = \frac{-\\sin \theta + \frac{1}{2} \\sin 2\theta}{\\sin \theta}\)

Simplify using the double angle identity \(\sin 2\theta = 2\\sin \theta \\cos \theta\):

\(\frac{dy}{dx} = \frac{-\\sin \theta + \\sin \theta \\cos \theta}{\\sin \theta} = -1 + \\cos \theta\)

Use the identity \(\\cos \theta = 1 - 2\\sin^2 \left( \frac{1}{2} \theta \right)\):

\(\frac{dy}{dx} = -1 + 1 - 2\\sin^2 \left( \frac{1}{2} \theta \right) = -2\\sin^2 \left( \frac{1}{2} \theta \right)\)