TITLE: Given that $\;\sin^3x\sin3x = \sum^n_{m=0}C_m\cos mx\,,\; C_n \neq 0\;$ is an identity . Find the value of n.
QUESTION [6 upvotes]: Problem : 
Given that $\sin^3 x \sin 3x = \sum^n_{m=0}C_m \cos mx, C_n \neq 0 $ is an identity. Find the value of n.
I tried : $\sin3x = 3\sin x - 4\sin^3 x$ but unable to reach to any point.... Please suggest further ....Thanks..

REPLY [6 votes]: Using the identity
$$
\sin^3(x)=\frac{3\sin(x)-\sin(3x)}{4}
$$
and the identity
$$
\sin(ax)\sin(bx)=\frac12\big(\cos((a-b)x)-\cos((a+b)x)\big)
$$
we get
$$
\begin{align}
\sin^3(x)\sin(3x)
&=\frac{3\sin(x)-\sin(3x)}{4}\sin(3x)\\
&=\frac38\big(\cos(2x)-\cos(4x)\big)-\frac18\big(\cos(0x)-\cos(6x)\big)\\
&=\frac18\big(\cos(6x)-3\cos(4x)+3\cos(2x)-\cos(0x)\big)
\end{align}
$$
Thus, $n=6$.