Write down the expansion of \(\sin 2 x\) in ascending powers of \(x\) up to and including the term in \(x ^ { 5 }\).
It is given that the first non-zero term in the expansion of
$$\sin 2 x - 2 x \left( 1 - p x ^ { 2 } \right) \left( 1 - x ^ { 2 } \right) ^ { - 1 }$$
in ascending powers of \(x\) is \(q x ^ { 5 }\).
Find the values of the rational numbers \(p\) and \(q\).