Question

Prove the following:
$\left(\sqrt{3}+1\right)\left(3-\cot {30}^{\circ }\right)={\tan }^3{60}^{\circ }-2\sin {60}^{\circ }$

Answer 1

$LHS=(\sqrt{3}+1)(3-\cot 30)=3\sqrt{3}-\sqrt{3}\cot 30+3-3\cot 30=3\sqrt{3}-\sqrt{3}\cot 30+\sqrt{3}\sqrt{3}-\sqrt{3}\sqrt{3}\cot 30=3\sqrt{3}-\sqrt{3}\cot 30(1+\sqrt{3})+3=3\sqrt{3}-\frac{\cot 30}{\tan 30}(1+\sqrt{3})+3=3\sqrt{3}-\frac{\cot 30}{\tan 30}\tan 45-\frac{\cot 30}{\tan 30\tan 30}+\cot {}^{2}30=\tan {}^{3}60-2\sin 60$