Suppose that \( \sin ^{3} x \sin 3 x=\sum_{m=0}^{n} c_{m} \cos m x \) is an identity in \( x \),...

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Suppose that \( \sin ^{3} x \sin 3 x=\sum_{m=0}^{n} c_{m} \cos m x \) is an identity in \( x \), where \( c_{0}, c_{1}, c_{2}, \ldots ., c_{n} \) are constants and \( c_{n} \neq 0 \), find the value of \( n \).
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