Prove that: $\sin ^{2} \frac{\pi}{18}+\sin ^{2} \frac{\pi}{9}+\sin ^{2} \frac{7 \pi}{18}+\sin ^{2} \frac{4 \pi}{9}=2$
$\mathrm{LHS}=\sin ^{2} \frac{\pi}{18}+\sin ^{2} \frac{\pi}{9}+\sin ^{2} \frac{7 \pi}{18}+\sin ^{2} \frac{4 \pi}{9}$
$=\sin ^{2} \frac{\pi}{18}+\sin ^{2} \frac{2 \pi}{18}+\sin ^{2} \frac{7 \pi}{18}+\sin ^{2} \frac{8 \pi}{18}$
$=\sin ^{2} \frac{\pi}{18}+\sin ^{2} \frac{2 \pi}{18}+\sin ^{2}\left(\frac{7 \pi}{18}\right)+\sin ^{2}\left(\frac{8 \pi}{18}\right)$
$=\sin ^{2} \frac{\pi}{18}+\sin ^{2} \frac{2 \pi}{18}+\sin ^{2}\left(\frac{\pi}{2}-\frac{2 \pi}{18}\right)+\sin ^{2}\left(\frac{\pi}{2}-\frac{\pi}{18}\right)$
$=\sin ^{2} \frac{\pi}{18}+\sin ^{2} \frac{2 \pi}{18}+\cos ^{2} \frac{2 \pi}{18}+\cos ^{2} \frac{\pi}{18}$
$=\sin ^{2} \frac{\pi}{18}+\cos ^{2} \frac{\pi}{18}+\sin ^{2} \frac{2 \pi}{18}+\cos ^{2} \frac{2 \pi}{18}$
$=1+1$
$=2$
= RHS
Hence, proved.
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