sin2 π/18 + sin2 π/9 + sin2 7π/18 + sin2 4π/9 =
(a) 1
(b) 4
(c) 2
(d) 0
(c) 2
We have:
$\sin ^{2} \frac{\pi}{18}+\sin ^{2} \frac{\pi}{9}+\sin ^{2} \frac{7 \pi}{18}+\sin ^{2} \frac{4 \pi}{9}$
$=\sin ^{2} \frac{\pi}{18}+\sin ^{2} \frac{2 \pi}{18}+\sin ^{2} \frac{7 \pi}{18}+\sin ^{2} \frac{8 \pi}{18}$
$=\sin ^{2} \frac{\pi}{18}+\sin ^{2} \frac{2 \pi}{18}+\sin ^{2}\left(\frac{7 \pi}{18}\right)+\sin ^{2}\left(\frac{8 \pi}{18}\right)$
$=\sin ^{2} \frac{\pi}{18}+\sin ^{2} \frac{2 \pi}{18}+\sin ^{2}\left(\frac{\pi}{2}-\frac{2 \pi}{18}\right)+\sin ^{2}\left(\frac{\pi}{2}-\frac{\pi}{18}\right)$
$=\sin ^{2} \frac{\pi}{18}+\sin ^{2} \frac{2 \pi}{18}+\cos ^{2} \frac{2 \pi}{18}+\cos ^{2} \frac{\pi}{18}$
$=\sin ^{2} \frac{\pi}{18}+\cos ^{2} \frac{\pi}{18}+\sin ^{2} \frac{2 \pi}{18}+\cos ^{2} \frac{2 \pi}{18}$
$=1+1$
= 2