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. 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..

HINT:
As you have noticed, $4\sin^3x=3\sin x-\sin3x,$
$$\text{So, }\sin^3x\sin3x=\frac{(3\sin x-\sin3x)\sin3x}4=\frac{3(2\sin x\sin3x)-2\sin^23x}8$$
As $2\sin A\sin B=\cos(A-B)-\cos(A+B),$
$\implies  2\sin x\sin3x=\cos(3x-x)-\cos(3x+x)=\cos2x-\cos4x$
and as $\cos2y=1-2\sin^2y\implies 2\sin^2y=1-\cos2y,$
$\implies 2\sin^23x=1-\cos6x$
