To find \(\frac{dy}{dx}\), we use the chain rule:

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

\(x = 3(1 + \sin^2 t)\)

\(\frac{dx}{dt} = 3 \cdot 2 \sin t \cdot \cos t = 6 \sin t \cos t\)

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

\(y = 2 \cos^3 t\)

\(\frac{dy}{dt} = 2 \cdot 3 \cos^2 t \cdot (-\sin t) = -6 \cos^2 t \sin t\)

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

\(\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{-6 \cos^2 t \sin t}{6 \sin t \cos t}\)

Simplify:

\(\frac{dy}{dx} = -\cos t\)