[tex]\bf \textit{quad-angle identities}\\\\\\
cos(4\theta)=8cos^4(\theta)-8cos^2(\theta)+1
\\\\\\
sin(4\theta)=
\begin{cases}
8sin(\theta)cos^3(\theta)-4sin(\theta)cos(\theta)\\\\
4sin(\theta)cos(\theta)-8sin^3(\theta)cos(\theta)
\end{cases}\\\\
-----------------------------\\\\
[/tex]
[tex]\bf sin(5\theta)\iff sin(4\theta + \theta)
\\\\\\
sin(4\theta)cos(\theta)+cos(4\theta)sin(\theta)
\\\\\\\
[4sin(\theta)cos(\theta)-8sin^3(\theta)cos(\theta)]cos(\theta)\\\\ + [8cos^4(\theta)-8cos^2(\theta)+1]sin(\theta)[/tex]
[tex]\bf -----------------------------\\\\
cos(5\theta)\iff cos(4\theta + \theta)
\\\\\\
cos(4\theta)cos(\theta)-sin(4\theta)sin(\theta)\\\\\\\
[8cos^4(\theta)-8cos^2(\theta)+1]cos(\theta)\\\\
-
[4sin(\theta)cos(\theta)-8sin^3(\theta)cos(\theta)]sin(\theta)[/tex]
distribute... and simplify :)
