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

MCQ

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

✓
$2 \cos \frac{A}{2} \cos \frac{B}{2} \sin \frac{ C }{2}$
B
$4 \cos \frac{A}{2} \cos \frac{B}{2} \sin \frac{ C }{2}$
C
$1-2 \cos \frac{A}{2} \cos \frac{B}{2} \sin \frac{ C }{2}$
D
$1-4 \cos \frac{A}{2} \cos \frac{B}{2} \sin \frac{ C }{2}$

✓

Answer

Correct option: A.

$2 \cos \frac{A}{2} \cos \frac{B}{2} \sin \frac{ C }{2}$

(A)
$\cos ^2 \frac{A}{2}+\cos ^2 \frac{B}{2}-\cos ^2 \frac{ C }{2}$
$=\frac{1}{2}(1+\cos A )+\frac{1}{2}(1+\cos B )-\frac{1}{2}(1+\cos C )$
$=\frac{1}{2}+\frac{1}{2}(\cos A+\cos B -\cos C )$
$\cos A +\cos B - \cos C =-1+4 \cos \frac{A}{2} \cos \frac{B}{2} \sin \frac{ C }{2}$
$=\frac{1}{2}+\frac{1}{2}\left[4 \cos \frac{A}{2} \cos \frac{B}{2} \sin \frac{ C }{2}-1\right]$
$=2 \cos \frac{A}{2} \cos \frac{B}{2} \sin \frac{ C }{2}$

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