(i) To find \(\frac{dy}{dx}\), we first find \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\).

\(x = \frac{1}{\cos^3 t} \Rightarrow \frac{dx}{dt} = \frac{3 \sin t}{\cos^4 t}\)

\(y = \tan^3 t \Rightarrow \frac{dy}{dt} = 3 \tan^2 t \sec^2 t\)

Now, \(\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{3 \tan^2 t \sec^2 t}{\frac{3 \sin t}{\cos^4 t}} = \sin t\)

(ii) The equation of the tangent to the curve at a point is given by \(y - y_1 = m(x - x_1)\), where \(m = \frac{dy}{dx}\).

At parameter \(t\), \(x_1 = \frac{1}{\cos^3 t}\) and \(y_1 = \tan^3 t\).

Substituting, \(y - \tan^3 t = \sin t \left(x - \frac{1}{\cos^3 t}\right)\)

Simplifying, \(y = x \sin t - \tan t\)