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If $4k = \tan^2 60^{\circ} - 2 \operatorname{cosec}^2 30^{\circ} -2 \tan^2 30^{\circ}$, then find the value of $k$.
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Sol. $4k = (\sqrt{3})^2 - 2(2)^2 - 2(\frac{1}{\sqrt{3}})^2$
$= 3 - 2(4) - 2(\frac{1}{3})$
$= 3 - 8 - \frac{2}{3}$
$= -5 - \frac{2}{3}$
$= \frac{-15-2}{3} = \frac{-17}{3}$
$k = \frac{-17}{12}$
$= 3 - 2(4) - 2(\frac{1}{3})$
$= 3 - 8 - \frac{2}{3}$
$= -5 - \frac{2}{3}$
$= \frac{-15-2}{3} = \frac{-17}{3}$
$k = \frac{-17}{12}$