The value of ${\cos }^4\frac{\pi }{8}+{\cos }^4\frac{3\pi }{8}+{\cos }^4\frac{5\pi }{8}+{\cos }^4\frac{7\pi }{8}$ is 

$\begin{array}{l}{\cos }^4\frac{\pi }{8}+{\cos }^4\frac{3\pi }{8}+{\cos }^4\frac{5\pi }{8}+{\cos }^4\frac{7\pi }{8}\\ =2\left({\cos }^4\frac{\pi }{8}+{\cos }^4\frac{3\pi }{8}\right)\\ =\frac{1}{2}\left[{\left(1+\cos \frac{\pi }{4}\right)}^2+{\left(1+\cos \frac{3\pi }{4}\right)}^2\right]\\ =\frac{1}{2}\left[{\left(1+\frac{1}{\sqrt{2}}\right)}^2+{\left(1-\frac{1}{\sqrt{2}}\right)}^2\right]=\frac{1}{2}\times 2\left(1+\frac{1}{2}\right)=\frac{3}{2}\end{array}$