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HURRY PLZ HELP EXAMS NEAR if A+B+C= pi prove that tan^2(A/2)+tan^2(B/2)+tan^2(C/2) is greater than or equal to 1

 HURRY PLZ HELP EXAMS NEAR
if A+B+C= pi
prove that tan^2(A/2)+tan^2(B/2)+tan^2(C/2) is greater than or equal to 1 

Grade:12

1 Answers

Ajay
209 Points
4 years ago
A/2 + B/2 = 90 – C/2
 
Tan (A/2 + B/2) = Tan (90-C/2) = Cot (C/2) = 1 /Tan C/2
(Tan A/2 + Tan B/2)/(1- Tan A/2*Tan B/2) = 1/Tan C/2
SImpify
Tan A/2 Tan B/2 + Tan A/2 Tan C/2 + Tan B/2 Tan C/2 = 1
We will use this  identty later.
 
Now since AM > GM
Implies Tan^2 A/2  + Tan^2 B/2 > 2 Tan A/2Tan B/2
             Tan^2 B/2  + Tan^2 C/2 > 2 Tan B/2Tan C/2
             Tan^2 A/2  + Tan^2 C/2 > 2 Tan A/2Tan C/2
 
Adding 
Tan^2 A/2  + Tan^2 B/2 + Tan ^2 C/2 >  Tan A/2Tan B/2 + Tan B/2Tan C/2 + Tan A/2Tan C/2
 
 But Tan A/2Tan B/2 + Tan B/2Tan C/2 + Tan A/2Tan C/2 = 1 as proved initailly we have 
Tan^2 A/2  + Tan^2 B/2 + Tan ^2 C/2 > 1

 

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