(i) Differentiate \(x = a \cos^3 t\) and \(y = a \sin^3 t\) with respect to \(t\):

\(\frac{dx}{dt} = -3a \cos^2 t \sin t\)

\(\frac{dy}{dt} = 3a \sin^2 t \cos t\)

Then, \(\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{3a \sin^2 t \cos t}{-3a \cos^2 t \sin t} = -\frac{\tan t}{\cos t} = -\tan t \sec t\).

(ii) The equation of the tangent is given by:

\(\frac{y - a \sin^3 t}{x - a \cos^3 t} = \frac{dy}{dx} = -\tan t \sec t\)

Rearranging gives:

\(x \sin t + y \cos t = a \sin t \cos t\).

(iii) The tangent meets the \(x\)-axis when \(y = 0\):

\(x \sin t = a \sin t \cos t \Rightarrow x = a \cos t\).

The tangent meets the \(y\)-axis when \(x = 0\):

\(y \cos t = a \sin t \cos t \Rightarrow y = a \sin t\).

The length \(XY\) is:

\(\sqrt{(a \cos t)^2 + (a \sin t)^2} = \sqrt{a^2 (\cos^2 t + \sin^2 t)} = a\).