Question

In a $\Delta ABC$, if $\cos A\cos B\cos C=\frac{\sqrt{3}-1}{8}$ and $\sin A.\sin B.\sin C=\frac{3+\sqrt{3}}{8}$,

then- On the basis of above information, answer the following questions: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 $5+4\sqrt{3}$

$A+B+C=\pi \Rightarrow \cos (A+B+C)=\cos \pi \Rightarrow \cos A\cos B\cos C-\sin A\sin B\cos C-\cos A\sin B\sin C-\sin A\cos B\sin C=-1\Rightarrow \cos A\cos B\cos C(1-\tan A\tan B-\tan B\tan C-\tan A\tan C)=-1\Rightarrow \frac{\sqrt{3}-1}{8}(1-\tan A\tan B-\tan B\tan C-\tan A\tan C)=-1\Rightarrow \tan A\tan B-\tan B\tan C-\tan A\tan C=5+4\sqrt{3}$