If tan A + tan B + tan C = tan A tan B tan C. Find the minimum value of tan A . tan B . tan C?

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Solution

The correct option is C

For tan A, tan B & tan C

AM $\geq$ GM

$\frac{t a n A + t a n B + t a n C}{3} \geq t a n A . t a n B . t a n C$

tan A . tan B . tan C $\geq$ 3 (tan A. tan B . tan C)

$(t a n A . t a n B . t a n C)^{\frac{2}{3}} \geq 3$

tan A . tan B . tan C $\geq (3)^{\frac{3}{2}}$

tan A . tan B . tan C $\geq 3 \sqrt{3}$

minimum value of tan A . tan B . tan C = $3 \sqrt{3}$

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