If A + B + C = , then ^2 A 2 + ^2 B 2 + ^2 C 2 is always - Vidyadip

MCQ

If $A + B + C = \pi ,$ then ${\tan ^2}\frac{A}{2} + {\tan ^2}\frac{B}{2} + $${\tan ^2}\frac{C}{2}$ is always

A
$ \le 1$
✓
$ \ge 1$
C
$= 0$
D
$= 1$

✓

Answer

Correct option: B.

$ \ge 1$

b
(b) $\tan \left( {\frac{A}{2} + \frac{B}{2} + \frac{C}{2}} \right) $

$= \frac{{{S_1} - {S_3}}}{{1 - {S_2}}} = \tan \frac{\pi }{2} = \infty $

$\therefore {S_2} = 1$ or $xy + yz + zx = 1$,

where $x = \tan \frac{A}{2}$etc.

Now ${(x - y)^2} + {(y - z)^2} + {(z - x)^2} \ge 0$

or $2\sum {x^2} - 2\sum xy \ge 0 \Rightarrow \sum {x^2} \ge 1$. $\{ \because \sum xy = 1\} $

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