Use De Moivre's Theorem to show that if \(z = \cos \theta + \mathrm { i } \sin \theta\), then
$$z ^ { n } - \frac { 1 } { z ^ { n } } = 2 \mathrm { i } \sin n \theta$$
Write down a similar expression for \(z ^ { n } + \frac { 1 } { z ^ { n } }\).
Expand \(\left( z - \frac { 1 } { z } \right) ^ { 2 } \left( z + \frac { 1 } { z } \right) ^ { 2 }\) in terms of \(z\).
Hence show that
$$8 \sin ^ { 2 } \theta \cos ^ { 2 } \theta = A + B \cos 4 \theta$$
where \(A\) and \(B\) are integers.
Hence, by means of the substitution \(x = 2 \sin \theta\), find the exact value of
$$\int _ { 1 } ^ { 2 } x ^ { 2 } \sqrt { 4 - x ^ { 2 } } \mathrm {~d} x$$
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