By considering the binomial expansion of \(\left( z + \frac { 1 } { z } \right) ^ { 7 }\) ,where \(z = \cos \theta + \mathrm { i } \sin \theta\) ,use de Moivre's theorem to show that
$$\cos ^ { 7 } \theta = a \cos 7 \theta + b \cos 5 \theta + c \cos 3 \theta + d \cos \theta$$
where \(a , b , c\) and \(d\) are constants to be determined.
Let \(I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \cos ^ { n } \theta \mathrm {~d} \theta\).
Show that
$$n I _ { n } = 2 ^ { - \frac { 1 } { 2 } n } + ( n - 1 ) I _ { n - 2 }$$
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Using the results given in parts (a) and (b), find the exact value of \(I _ { 9 }\).
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