First, find \(\\frac{dx}{dt}\):

\(\\frac{dx}{dt} = \\frac{d}{dt}(2t - \\tan t) = 2 - \\sec^2 t\).

Next, find \(\\frac{dy}{dt}\):

\(\\frac{dy}{dt} = \\frac{d}{dt}(\\ln(\\sin 2t)) = \\frac{1}{\\sin 2t} \\cdot \\frac{d}{dt}(\\sin 2t) = \\frac{1}{\\sin 2t} \\cdot 2 \\cos 2t = \\frac{2 \\cos 2t}{\\sin 2t}\).

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

\(\\frac{dy}{dx} = \\frac{dy}{dt} \\div \\frac{dx}{dt} = \\frac{2 \\cos 2t}{\\sin 2t} \\div (2 - \\sec^2 t)\).

Simplify using trigonometric identities:

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

Using the double angle formula, \(\\sin 2t = 2 \\sin t \\cos t\), and \(\\cos 2t = \\cos^2 t - \\sin^2 t\), we simplify:

\(\\frac{dy}{dx} = \\frac{\\cos t}{\\sin t} = \\cot t\).