Animated Solution for Mathematics - Trigonometry: Suppose ${\sin }^{3}x\sin 3x={\sum }_{m=0}^n{C}_m\cos mx$ is an identity in $x$, where $C_0,C_1,\dots ,C_n$ are constants, and $C_n\ne 0$. then the value of $n$ is ..........

The objective is to transform the expression ${\sin }^{3}x\sin 3x$ into a linear summation of cosine terms of the form ${\sum }_{m=0}^n{C}_m\cos mx$. Our primary goal is to determine the value of $n$, which represents the highest frequency (the maximum coefficient of $x$) present in the resulting identity.

Breaking the Power

The term ${\sin }^{3}x$ is a power that complicates the summation. To linearize it, we utilize the triple angle identity:

By rearranging this identity to isolate ${\sin }^{3}x$, we obtain:

This yields the linearized form:

The Product and the Square

Substituting this back into our original expression, we have:

Distributing $\sin 3x$ across the terms, we get:

To resolve the product $3\sin x\sin 3x$, we use the product-to-sum identity $2\sin A\sin B=\cos (A-B)-\cos (A+B)$. For the squared term ${\sin }^{2}3x$, we apply the power reduction identity ${\sin }^2\theta =\frac{1-\cos 2\theta }{2}$.

The Final Assembly

Applying the product-to-sum formula to the first term:

Applying the power reduction formula to the second term with $\theta =3x$:

Combining these results into the main expression:

Factoring out $\frac{1}{2}$ from the bracketed terms, we arrive at the final expansion:

The expression is now represented as a linear sum of cosines:

Comparing this to the form ${\sum }_{m=0}^n{C}_m\cos mx$, we identify that the highest multiple of $x$ is $6$. Therefore, the value of $n=6$.