Problem

The curve $C$ has parametric equations

$$x = 8\ln\big(\tan \tfrac{1}{2}t\big) - 3\cot t - 3t, \quad y = 6\ln\big(\tan \tfrac{1}{2}t\big) + 4\cot t + 4t,$$

for $\frac{1}{6}\pi \le t \le \frac{1}{3}\pi$.

(a) (i) Show that $\frac{\mathrm{d}x}{\mathrm{d}t} = 8\cosec t + 3\cot^2 t$.

(ii) Find $\frac{\mathrm{d}y}{\mathrm{d}t}$.

(iii) Find the exact value of the length of $C$.

(b) Show that

$$\frac{\mathrm{d}^2y}{\mathrm{d}x^2} = \frac{ a\cot t\cosec t(\cosec^2 t + 1) }{ \big(b\cosec t + c\cot^2 t\big)^n },$$

where $a$, $b$, $c$ and $n$ are integers to be determined.