Question

The value of $\tan A\tan B+\tan B\tan C+\tan C\tan A$ is

• A. $5-4\sqrt{3}$
• B. $5+4\sqrt{3}$
• C. $6+\sqrt{3}$
• D. $6-\sqrt{3}$

Answer 1

The correct option is B $5+4\sqrt{3}$

$\because \cos A\cos B\cos C=\frac{\sqrt{3}-1}{8}$ and $\sin A\sin B\sin C=\frac{3+\sqrt{3}}{8}$

$\therefore \tan A\tan B\tan C=\frac{3+\sqrt{3}}{\sqrt{3}-1}$

As in $△ABC,A+B+C=\pi$

$\therefore \tan A\tan B\tan C=\tan A+\tan B+\tan C=\frac{3+\sqrt{3}}{\sqrt{3}-1}$

Now $A+B+C=\pi \Rightarrow \cos (A+B+C)=-1$

$\Rightarrow \cos A\cos B\cos C[1-\sum \tan A\tan B]=-1$

$\Rightarrow \frac{\sqrt{3}-1}{8}[1-\sum \tan A\tan B]=-1$

$\Rightarrow \sum \tan A\tan B=5+4\sqrt{3}$