Prove that $\cot ^{2} \frac{\pi}{6}+\operatorname{cosec} \frac{5 \pi}{6}+3 \tan ^{2} \frac{\pi}{6}=6$

solution:

L.H.S. $=\cot ^{2} \frac{\pi}{6}+\operatorname{cosec} \frac{5 \pi}{6}+3 \tan ^{2} \frac{\pi}{6}$

$=(\sqrt{3})^{2}+\operatorname{cosec}\left(\pi-\frac{\pi}{6}\right)+3\left(\frac{1}{\sqrt{3}}\right)^{2}$

$=3+\operatorname{cosec} \frac{\pi}{6}+3 \times \frac{1}{3}$

$=3+2+1=6$

$=$ R.H.S