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If $4k = \tan^2 60^\circ - 2 \text{cosec}^2 30^\circ - 2 \tan^2 30^\circ$, then find the value of $k$.
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$$\begin{aligned}& 4k = (\sqrt{3})^2 - 2(2)^2 - 2\left(\frac{1}{\sqrt{3}}\right)^2 \\ & = -\frac{17}{3} \\ & k = -\frac{17}{12}\end{aligned}$$