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

\(\frac{dx}{d\theta} = \frac{d}{d\theta}(\sin 2\theta - \theta) = 2\cos 2\theta - 1\).

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

Now, use the chain rule to find \(\frac{dy}{dx}\):

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

Using the double angle identities, \(\sin 2\theta = 2\sin \theta \cos \theta\) and \(\cos 2\theta = 1 - 2\sin^2 \theta\), substitute:

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

Simplify the expression:

\(\frac{dy}{dx} = \frac{-4\sin \theta \cos \theta + 2\cos \theta}{2 - 4\sin^2 \theta - 1}\).

\(\frac{dy}{dx} = \frac{2\cos \theta(1 - 2\sin \theta)}{1 - 2\sin^2 \theta}\).

Further simplify using \(1 - 2\sin^2 \theta = \cos 2\theta\):

\(\frac{dy}{dx} = \frac{2\cos \theta}{1 + 2\sin \theta}\).