Ajay
Last Activity: 8 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
