Problem

The curve $C$ has parametric equations

$$x=3+2\sin t \qquad y=\frac{6}{7+\cos 2t}\qquad -\frac{\pi}{2}\leq t\leq \frac{\pi}{2}$$

(a) Show that $C$ has Cartesian equation

$$y=\frac{12}{(7-x)(1+x)} \qquad p\leq x\leq q$$

where $p$ and $q$ are constants to be found.

(b) Hence, find a Cartesian equation for $C$ in the form

$$y=\frac{a}{x+b}+\frac{c}{x+d}\qquad p\leq x\leq q$$

where $a$, $b$, $c$ and $d$ are constants.