First slide

Multiple and sub- multiple Angles

Question

$In Δ A B C \cos A \cos B \cos C = \frac{\sqrt{3} - 1}{8} and \sin A \sin B \sin C = \frac{3 + \sqrt{3}}{8}$ , then the value of $\tan A + \tan B + \tan C =$

Moderate

A

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

$\frac{\sqrt{3} + 4}{\sqrt{3} - 1}$
C

$\frac{6 - \sqrt{3}}{\sqrt{3} - 1}$
D

$\frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - 1}$

Solution

We have
$\begin{array}{l} \frac{\sin A \sin B \sin C}{\cos A \cos B \cos C} = \frac{3 + \sqrt{3}}{\sqrt{3} - 1} \\ \Rightarrow \tan A \tan B \tan C = \frac{3 + \sqrt{3}}{\sqrt{3} - 1} \end{array} \Rightarrow \tan A + \tan B + \tan C = \frac{3 + \sqrt{3}}{\sqrt{3} - 1} (∵ A + B + C = π \Rightarrow tanA + tanB + tanC = tanAtanBtanC)$

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