Show that, if \(y = \operatorname { cosec } x\), then \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) can be expressed as \(- \operatorname { cosec } x \cot x\).
Solve the differential equation
$$\frac { \mathrm { d } x } { \mathrm {~d} t } = - \sin x \tan x \cot t$$
given that \(x = \frac { 1 } { 6 } \pi\) when \(t = \frac { 1 } { 2 } \pi\).