Question

Cot30÷1-tan30+tan30÷1-cot30

Answer 1

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Answer 2

$The \: value \: of \:\frac{Cot 30}{1 - tan 30} + \frac{ tan 30}{1 - cot 30 }$

$We \: know \:that ,\\</p><p>tan 30\degree = \frac{1}{\sqrt{3}},\\cot 30\degree = \sqrt{3}$

$= \frac{\sqrt{3}}{1 - \frac{1}{\sqrt{3}}} + \frac{\frac{1}{\sqrt{3}}}{ 1 - \sqrt{3}}$

$= \frac{\sqrt{3}}{ \frac{\sqrt{3} - 1}{\sqrt{3}}} + \frac{1}{\sqrt{3} ( 1- \sqrt{3})}$

$= \frac{\sqrt{3}\times \sqrt{3}}{(\sqrt{3} - 1)} - \frac{1}{\sqrt{3}(\sqrt{3} - 1)}$

$= \frac{3}{(\sqrt{3} - 1)} - \frac{1}{\sqrt{3}(\sqrt{3} - 1)}$

$= \frac{3\sqrt{3} - 1}{\sqrt{3}(\sqrt{3} - 1)}$

$= \frac{(3\sqrt{3} - 1)(\sqrt{3}+1)}{\sqrt{3}(\sqrt{3} +1)(\sqrt{3}+1)}$

$= \frac{3\sqrt{3}\times \sqrt{3} + 3\sqrt{3} - \sqrt{3} - 1 }{ \sqrt{3} \times ( \sqrt{3})^{2} - 1^{2})}$

$= \frac{ 9 + 3\sqrt{3} - \sqrt{3} + 1}{\sqrt{3} ( 3-1)}$

$= \frac{ 10 + 2\sqrt{3}}{2\sqrt{3}}\\= \frac{2(5+\sqrt{3})}{2\sqrt{3}}\\= \frac{5+\sqrt{3}}{2\sqrt{3}}$

Therefore.,

$\red{The \: value \: of \:\frac{Cot 30}{1 - tan 30} + \frac{ tan 30}{1 - cot 30 }}$

$\green { = \frac{5+\sqrt{3}}{2\sqrt{3}}}$

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