By differentiating \(\frac { \cos x } { \sin x }\), show that if \(y = \cot x\) then \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \operatorname { cosec } ^ { 2 } x\).
Hence show that \(\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 2 } \pi } \operatorname { cosec } ^ { 2 } x \mathrm {~d} x = \sqrt { } 3\).
By using appropriate trigonometrical identities, find the exact value of