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  4. cot 2 (π/6)+ operatornamecosec (5 π/6)+3 tan 2 (π/6) is equal to

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Q. $\cot ^2 \frac{\pi}{6}+\operatorname{cosec} \frac{5 \pi}{6}+3 \tan ^2 \frac{\pi}{6}$ is equal to

Trigonometric Functions

Solution:

$\cot ^2 \frac{\pi}{6}+\operatorname{cosec} \frac{5 \pi}{6}+3 \tan ^2 \frac{\pi}{6}$
$=(\sqrt{3})^2+\operatorname{cosec}\left(\pi-\frac{\pi}{6}\right)+3\left(\frac{1}{\sqrt{3}}\right)^2$
$=3+\operatorname{cosec} \frac{\pi}{6}+3 \times \frac{1}{3} [\because \operatorname{cosec}(\pi-\theta)=\operatorname{cosec} \theta]$
$=3+2+1=6$