Question

Find the expansions of sin⁡5θ and cos⁡5θ in terms of sin⁡θ and cos⁡θ.

Answer 1

$\bf \textit{quad-angle identities}\\\\\\ cos(4\theta)=8cos^4(\theta)-8cos^2(\theta)+1 \\\\\\ sin(4\theta)= \begin{cases} 8sin(\theta)cos^3(\theta)-4sin(\theta)cos(\theta)\\\\ 4sin(\theta)cos(\theta)-8sin^3(\theta)cos(\theta) \end{cases}\\\\ -----------------------------$

$\bf sin(5\theta)\iff sin(4\theta + \theta) \\\\\\ sin(4\theta)cos(\theta)+cos(4\theta)sin(\theta) \\\\\\\ [4sin(\theta)cos(\theta)-8sin^3(\theta)cos(\theta)]cos(\theta)\\\\ + [8cos^4(\theta)-8cos^2(\theta)+1]sin(\theta)$

$\bf -----------------------------\\\\ cos(5\theta)\iff cos(4\theta + \theta) \\\\\\ cos(4\theta)cos(\theta)-sin(4\theta)sin(\theta)\\\\\\\ [8cos^4(\theta)-8cos^2(\theta)+1]cos(\theta)\\\\ - [4sin(\theta)cos(\theta)-8sin^3(\theta)cos(\theta)]sin(\theta)$

distribute... and simplify :)