$\cos \theta-\sin \theta=\sqrt{5} \sin \theta$
$\begin{aligned} & (\cos \theta-\sin \theta)^2=(\sqrt{5} \sin \theta)^2 \\ & \cos ^2 \theta+\sin ^2 \theta-2 \sin \theta \cos \theta=5 \sin ^2 \theta \\ & 1-2 \sin \theta \cos \theta=5 \sin ^2 \theta \\ & 1-5 \sin ^2 \theta=2 \sin \theta \cos \theta\end{aligned}$
$Now,$
$(\cos \theta+4 \sin \theta)^2=\cos ^2 \theta+16 \sin ^2 \theta+4.2 \sin \theta \cos \theta$
$\begin{aligned} & =\cos ^2 \theta+16 \sin ^2 \theta+\left(1-5 \sin ^2 \theta\right) \\ & =\cos ^2 \theta+16 \sin ^2 \theta+4-20 \sin ^2 \theta \\ & =\cos ^2 \theta-4 \sin ^2 \theta+4 \\ & =\cos ^2 \theta+4\left(1-\sin ^2 \theta\right) \\ & =\cos ^2 \theta+4 \cos ^2 \theta \\ & =5 \cos ^2 \theta\end{aligned}$
$\cos \theta+4 \sin \theta=\sqrt{5 \cos ^2 \theta}$
$=\sqrt{5} \cdot \cos \theta$