Question

In an actue-angled triangle ABC
Statement-1: `tan^(2)""(A)/(2)+tan^(2)""(B)/(2)+tan^(2)""(C)/(2)ge1`
Statement-2: `tanAtanB tanCge3sqrt3`
A. Statement-1 is True, Statement-2 is true, Statement-2 is a correct explanation for Statement -1.
B. Statement-1 is True, Statement-2 is True, Statement-2 is not a correct explanation for Statement-1.
C. Statement-1 is True, Statement-2 is False.
D. Statement-1 is False, Statement-2 is True.

Answer 1

Correct Answer - B
We have, `A+B+C=pi`
`thereforetan(A+B+C)=tanpiand tan((A)/(2)+(B)/(2)+(C)/(2))=tan""(pi)/(2)`
`implies(S_(1)-S_(3))/(1-S_(3))=0and(1-S_(2))/(S_(1)-S_(3))=0`
`impliesS_(1)=S_(3)and S_(2)=1`
`impliestanA+tanB+tanC=tanAtanBtanC" "...(i)`
`and, tan""(A)/(2)tan""(B)/(2)+tan""(B)/(2)tan""(C)/(2)+tan""(C)/(2)tan""(A)/(2)=1" "...(ii)`
Using `A.M. geG.M.,` we have
`(tanAtanB+tanC)/(3)ge(tanAtanBtanC)^(1//3)`
`implies(tanAtanBtanC)/(3)ge(tanA tanBtanC)^(1//3)" "["Using"(i)]`
`impliestanA tanBtanCge3sqrt3`
So, statement-2 is true.
From (ii), we have
`xy+yz+zx=1,where x=tan""(A)/(2),y=tan""(B)/(2)and z=tan""(C)/(2)`
`thereforex^(2)+y^(2)+z^(2)-1`
`=x^(2)+y^(2)+z^(2)-(xy+yz+zx)`
`=1/2{(x-y)^(2)+(y-z)^(2)+(z-y)^(2)}ge0`
`impliesx^(2)+y^(2)+z^(2)gt1impliestan^(2)""(A)/(2)+tan^(2)""(B)/(2)+tan^(2)""(C)/(2)ge1`
Hence, both the statements are true.