Given that \(y = \frac { \sin \theta } { \cos \theta }\), use the quotient rule to show that \(\frac { \mathrm { d } y } { \mathrm {~d} \theta } = \sec ^ { 2 } \theta\).
Given that \(x = \sin \theta\), show that \(\frac { x } { \sqrt { 1 - x ^ { 2 } } } = \tan \theta\).
Use the substitution \(x = \sin \theta\) to find \(\int \frac { 1 } { \left( 1 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x\), giving your answer in terms of \(x\).