9. A curve has parametric equations
$$x = \sec \theta + \tan \theta , \quad y = \operatorname { cosec } \theta + \cot \theta , \quad 0 < \theta < \frac { \pi } { 2 }$$
- Show that \(x + \frac { 1 } { x } = 2 \sec \theta\).
Given that \(y + \frac { 1 } { y } = 2 \operatorname { cosec } \theta\),
- find a cartesian equation for the curve.
- Show that \(\frac { \mathrm { d } x } { \mathrm {~d} \theta } = \frac { 1 } { 2 } \left( x ^ { 2 } + 1 \right)\).
- Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).