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,  \( \left( \sqrt{3}+1\right) \left(3- \cot 30^{\omicron}\right)=\tan^3 60^{\omicron}-2\sin 60^{\omicron}\)