(i) To differentiate \(\frac{1}{\sin^2 \theta}\), rewrite it as \(\csc^2 \theta\). The derivative of \(\csc \theta\) is \(-\csc \theta \cot \theta\). Using the chain rule, the derivative of \(\csc^2 \theta\) is:

\(-2 \csc \theta \cot \theta \cdot \csc \theta = -2 \csc^2 \theta \cot \theta\).

(ii) Start with the differential equation:

\(x \tan \theta \frac{dx}{d\theta} + \csc^2 \theta = 0\).

Rearrange to separate variables:

\(x \tan \theta \frac{dx}{d\theta} = -\csc^2 \theta\).

\(x \frac{dx}{d\theta} = -\frac{\csc^2 \theta}{\tan \theta}\).

Integrate both sides:

\(\int x \, dx = \int -\csc^2 \theta \cot \theta \, d\theta\).

\(\frac{1}{2}x^2 = \frac{1}{\sin^2 \theta} + C\).

Use the initial condition \(x = 4\) when \(\theta = \frac{1}{6}\pi\):

\(\frac{1}{2}(4)^2 = \frac{1}{\sin^2 \frac{1}{6}\pi} + C\).

\(8 = 4 + C\).

\(C = 4\).

Thus, the solution is:

\(\frac{1}{2}x^2 = \frac{1}{\sin^2 \theta} + 4\).

\(x^2 = \frac{2}{\sin^2 \theta} + 8\).

\(x = \sqrt{\csc^2 \theta + 12}\).