In a `DeltaABC, "the value of"cot""(A)/(2)cot""(B)/(2)cot""(C)/(2)is`

Options

A. `ge3sqrt3`
B. `ge9`
C. `ge6sqrt3`
D. none of these

Answer

A

Explanation

In `DeltaABC,` we have
`tan""(A)/(2)tan""(B)/(2)+tan""(B)/(2)tan""(C)/(2)+tan""(C)/(2)tan""(A)/(2)=1`
`impliescot""(A)/(2)tan""(B)/(2)+cot""(C)/(2)=cot""(A)/(2)cot""(B)/(2)cot""(C)/(2)`
`Now, " "A.M.geG.M.`
`implies{(cot""(A)/(2)+cot""(B)/(2)+cot""(C)/(2))/(3)}ge{cot""(A)/(2)cot""(B)/(2)cot""(C)/(2)}^(1//3)`
`implies{(cot""(A)/(2)cot""(B)/(2)cot""(C)/(2))/(3)}ge{cot""(A)/(2)cot""(B)/(2)cot""(C)/(2)}^(1//3)`
`implies{cot""(A)/(2)cot""(B)/(2)cot""(C)/(2)}^(2//3)ge3`
`impliescot""(A)/(2)cot""(B)/(2)cot""(C)/(2)ge3sqrt3.`