[tex]\bf \qquad \qquad \textit{triple-angle identity}\\\\
sin(3\theta)=3sin(\theta)-4sin^3(\theta)\\\\
\left. \qquad \qquad \right.\textit{quad-angle identity}\\\\
sin(4\theta)=
\begin{cases}
8sin(\theta)cos^3(\theta)=4sin(\theta)cos(\theta)\\\\
\boxed{4sin(\theta)cos(\theta)-8sin^3(\theta)cos(\theta)}
\end{cases}\\\\
-----------------------------\\\\
2cos(x)[sin(3x)-sin(x)]=sin(4x)\\\\
-----------------------------\\\\[/tex]
[tex]\bf 2cos(x)[sin(3x)-sin(x)]\implies 2cos(x)[3sin(x)-4sin^3(x)-sin(x)]
\\\\\\
2cos(x)[2sin(x)-4sin^3(x)]\implies \boxed{4sin(x)cos(x)-8sin^3(x)cos(x)}[/tex]
