Mathematics - Trigonometric Ratios & Identities Question with Solution | TestHub

Since $A+B+C=\pi ,\tan A+\tan B+\tan C=\tan A\tan B\tan C$. As the triangle is acute, all tangents are positive, so by A.M. $\ge$ G.M., $\tan A\tan B\tan C=\tan A+\tan B+\tan C\ge 3(\tan A\tan B\tan C{)}^{1/3}$, giving $\tan A\tan B\tan C\ge 3\sqrt{3}$. Again by A.M. $\ge$ G.M., $\sum \tan A\tan B\ge 3(\tan A\tan B\tan C{)}^{2/3}\ge 3(3\sqrt{3}{)}^{2/3}=9$. Equality holds for the equilateral triangle, so the minimum value is 9.