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.

In an actue-angled triangle ABC Statement-1: `tan^(2)""(A)/(2)+tan^(2)

1.	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» 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.