Question

If A, B, C are actual positive angles such that A + B + C = $\pi$ and cot A cot B cot C = k then

• A.
• B.
• C.
• D.

Answer 1

The correct option is A

A + B + C =$\pi$

A + B = $\pi$ - C

tan (A + B = tan ($\pi$ - C)

$\frac{tanA+tanB}{1-tanA.tanB}$ = -tanC

tan A + tan B + tan C = tan A tan B tan C - - - - - - (1)

For three values tan A, tan B & tan C

AM $\ge$ GM

$\frac{tanA+tanB+tanC}{3}$ $\ge$ $(tanA.tanB.tanC{)}^{\frac{1}{3}}$

[tanA + tanB + tanC = tanA tanB tanC]

$(tanA.tanB.tanC{)}^{\frac{2}{3}}\ge 3$

${\left(\frac{1}{cotA.cotB.cotC}\right)}^{\frac{2}{3}}\ge 3$

{cotA.cotB.cotC = k}

${\left(\frac{1}{k}\right)}^{\frac{2}{3}}\ge 3$

$\frac{1}{k}\ge (3{)}^{\frac{3}{2}}$

k $\le \frac{1}{(3{)}^{\frac{3}{2}}}$

k $\le \frac{1}{{3.3}^{\frac{1}{2}}}$

k $\le \frac{1}{3\sqrt{3}}$