Given that
$$x = \ln ( \sec t + \tan t ) - \sin t$$
show that
$$\frac { \mathrm { d } x } { \mathrm {~d} t } = \sin t \tan t$$
A curve is given parametrically by the equations
$$x = \ln ( \sec t + \tan t ) - \sin t , \quad y = \cos t$$
The length of the arc of the curve between the points where \(t = 0\) and \(t = \frac { \pi } { 3 }\) is denoted by \(s\).
Show that \(s = \ln p\), where \(p\) is an integer.