If \(\frac{k\;-\;k\;{cot}^{2}30°}{1\;+\;{cot}^{2}30°}\)= sin²60° + 4 tan²45° - cosec²60°, then find the value of k.

-6.83

⇒ \(\frac{k\;(1 - {cot}^{2}30°)}{1 + {cot}^{2}30°}\) = \( {(\frac{\sqrt {3}}{2})}^{2}\) + 4 × 1- (\(\frac{2}{\sqrt {3}})^2\)

⇒ \(\frac{k\;(1 - 3)}{1 + 3}\) = \(\frac{3}{4}\) + 4  - \(\frac{4}{3}\)

⇒ \(\frac{-2k}{4}\) = \(\frac{3}{4}\) + 4 - \(\frac{4}{3}\)

⇒ k = (\(\frac{9 + 48 - 16}{12}\)) × -2

⇒ k = \(\frac{-41}{6}\)

⇒ k = -6.83