Prove that:

$\mathrm{LHS}=\cos ^{2} \frac{\pi}{8}+\cos ^{2} \frac{3 \pi}{8}+\cos ^{2} \frac{5 \pi}{8}+\cos ^{2} \frac{7 \pi}{8}$

$=\cos ^{2} \frac{\pi}{8}+\cos ^{2} \frac{3 \pi}{8}+\cos ^{2}\left(\pi-\frac{3 \pi}{8}\right)+\cos ^{2}\left(\pi-\frac{\pi}{8}\right)$

$=\cos ^{2} \frac{\pi}{8}+\cos ^{2} \frac{3 \pi}{8}+\left\{-\cos \left(\frac{3 \pi}{8}\right)\right\}^{2}+\left\{-\cos \left(\frac{\pi}{8}\right)\right\}^{2}$

$=\cos ^{2} \frac{\pi}{8}+\cos ^{2} \frac{3 \pi}{8}+\cos ^{2} \frac{3 \pi}{8}+\cos ^{2} \frac{\pi}{8}$

$=2\left(\cos ^{2} \frac{\pi}{8}+\cos ^{2} \frac{3 \pi}{8}\right)$

$=2\left\{\cos ^{2} \frac{\pi}{8}+\cos ^{2}\left(\frac{\pi}{2}-\frac{\pi}{8}\right)\right\}$

$=2\left(\cos ^{2} \frac{\pi}{8}+\sin ^{2} \frac{\pi}{8}\right)$

$=2=$ RHS

Hence proved.