Question

Prove that
$(\sqrt{3} + 1)(3 - \cot(30)) = \tan^{3} (60) - 2 \tan(60)$

Answer 1

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$(\sqrt{3} + 1)(3 - \cot(30)) = \tan^{3} (60) - \tan(60) \\ \\ lhs. \\ = (\sqrt{3} + 1)(3 - \cot(30)) \\ \\ = (\sqrt{3} + 1)(3 - \sqrt{3} ) \\ \\ = 3 \sqrt{3} - 3 + 3 - \sqrt{3} \\ \\ = 2 \sqrt{3} \\ \\ rhs. \\ = \tan^{3} (60) - \tan(60) \\ \\ = (\sqrt{3} )^{3} - \sqrt{3} \\ \\ = 3 \sqrt{3} - \sqrt{3} \\ \\ = 2\sqrt{3} \\ \\ :. \: rhs = lhs$
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