Prove that:

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

$=\sin ^{2}\left(\frac{\pi}{2}-\frac{3 \pi}{8}\right)+\sin ^{2}\left(\frac{\pi}{2}-\frac{\pi}{8}\right)+\sin ^{2} \frac{5 \pi}{8}+\sin ^{2} \frac{7 \pi}{8}$

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

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

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

$=1+1=2=$ RHS

Hence proved.