To Prove  \( \left( \sqrt{3}+1\right) \left(3- \cot 30^{\omicron}\right)=\tan^3 60^{\omicron}-2\sin 60^{\omicron}\) Solution: Step 1: Evaluate the Value of L.H.S \( \left( \sqrt{3}+1\right) \left(3- \cot 30^{\omicron}\right)\) \( =\left( \sqrt{3}+1\right) \left(3- \sqrt{3}\right)\) \( =\left( 3\sqrt{3}- \sqrt{3}\times \sqrt{3}+1\times 3-1\times \sqrt{3}\right) \) \( =\left( 3\sqrt{3}-3+3-\sqrt{3}\right) \) \( = 2\sqrt{3} \) Step 2: Evaluate the Value of R.H.S \(\tan^3 60^{\omicron}-2\sin 60^{\omicron}\) \(= \left(\sqrt{3}\right)^3-2 \times \frac{\sqrt{3}}{2}\) \(= \left(\sqrt{3}\right)^2\times \sqrt{3}-\sqrt{3}\) \(= 3\times \sqrt{3}-\sqrt{3}\) \(= 3 \sqrt{3}-\sqrt{3}\) \(=2 \sqrt{3}\) As both L.H.S = R.H.S Therefore,  \( \left( \sqrt{3}+1\right) \left(3- \cot 30^{\omicron}\right)=\tan^3 60^{\omicron}-2\sin 60^{\omicron}\)