The $n^{th}$derivative of ${\cos }^4 \mathrm{x}$ is

$\begin{array}{l}\mathrm{y}={\cos }^4\mathrm{x}={\left({\cos }^2\mathrm{x}\right)}^2={\left(\frac{1+\cos 2\mathrm{x}}{2}\right)}^2=\frac{1+2\cos 2\mathrm{x}+{\cos }^{2}2\mathrm{x}}{4}=\frac{1}{4}+\frac{1}{2}\cos 2\mathrm{x}+\frac{1+\cos 4\mathrm{x}}{8}\\ =\frac{3}{8}+\frac{1}{2}\cos 2\mathrm{x}+\frac{1}{8}\cos 4\mathrm{x}\\ \therefore {\mathrm{y}}_{\mathrm{n}}=\frac{1}{2}\cdot 2^{\mathrm{n}}\cos \left(2\mathrm{x}+\frac{\mathrm{n\pi }}{2}\right)+\frac{1}{8}\cdot 4^{\mathrm{n}}\cos \left(4\mathrm{x}+\frac{\mathrm{n\pi }}{2}\right)=2^{\mathrm{n}-1}\cos \left(2\mathrm{x}+\frac{\mathrm{n\pi }}{2}\right)+2^{2\mathrm{n}-3}\cos \left(4\mathrm{x}+\frac{\mathrm{n\pi }}{2}\right)\end{array}$