$\cos 5 \theta=\cos (4 \theta+\theta)$
$\begin{aligned} & =\cos 4 \theta \cdot \cos \theta-\sin 4 \theta \sin \theta \\ & =\left(2 \cos ^2 2 \theta-1\right) \cos \theta-2 \sin 2 \theta \cos 2 \theta \sin \theta \\ & =\left\{2\left(2 \cos ^2 \theta-1\right)^2-1\right\} \cos \theta-4 \sin ^2 \theta \cos \theta \cos 2 \theta\end{aligned}$
$\begin{array}{r}=\left(8 \cos ^4 \theta+1-8 \cos ^2 \theta\right) \cos \theta-4\left(1-\cos ^2 \theta\right) \cos \theta \\ \left(2 \cos ^2 \theta-1\right)\end{array}$
$\begin{aligned} & =\left(8 \cos ^4 \theta+1-8 \cos ^2 \theta\right) \cos \theta-4 \cos \theta\left(3 \cos ^2 \theta-1-2 \cos ^4 \theta\right) \\ & =8 \cos ^5 \theta+\cos \theta-8 \cos ^3 \theta-12 \cos ^3 \theta+4 \cos \theta+8 \cos ^5 \theta \\ & =16 \cos ^5 \theta-20 \cos ^3 \theta+5 \cos \theta\end{aligned}$