Prove that $2 \sin ^{2} \frac{\pi}{6}+\operatorname{cosec}^{2} \frac{7 \pi}{6} \cos ^{2} \frac{\pi}{3}=\frac{3}{2}$

solution:

L.H.S. $=2 \sin ^{2} \frac{\pi}{6}+\operatorname{cosec}^{2} \frac{7 \pi}{6} \cos ^{2} \frac{\pi}{3}$

$=2\left(\frac{1}{2}\right)^{2}+\operatorname{cosec}^{2}\left(\pi+\frac{\pi}{6}\right)\left(\frac{1}{2}\right)^{2}$

$=2 \times \frac{1}{4}+\left(-\operatorname{cosec} \frac{\pi}{6}\right)^{2}\left(\frac{1}{4}\right)$

$=\frac{1}{2}+(-2)^{2}\left(\frac{1}{4}\right)$

$=\frac{1}{2}+\frac{4}{4}=\frac{1}{2}+1=\frac{3}{2}$

$=$ R.H.S.