If A + B + C =180^ , then the value of A 2 + B 2 + C 2 will be - Vidyadip

MCQ

If $A + B + C =180^{\circ}$, then the value of $\cot \frac{ A }{2}+\cot \frac{ B }{2}+\cot \frac{ C }{2}$ will be

A
$2 \cot \frac{A}{2} \cot \frac{B}{2} \cot \frac{ C }{2}$
B
$4 \cot \frac{A}{2} \cot \frac{B}{2} \cot \frac{ C }{2}$
✓
$\cot \frac{ A }{2} \cot \frac{B}{2} \cot \frac{ C }{2}$
D
$8 \cot \frac{A}{2} \cot \frac{B}{2} \cot \frac{ C }{2}$

✓

Answer

Correct option: C.

$\cot \frac{ A }{2} \cot \frac{B}{2} \cot \frac{ C }{2}$

(C)
$A+B+C=180^{\circ}$
$\Rightarrow \cot \left(\frac{ A }{2}+\frac{ B }{2}\right)=\cot \left(90^{\circ}-\frac{ C }{2}\right)$
$\Rightarrow \frac{\cot _2^{ A } \cdot \cot _2^{ B }-1}{\cot \frac{ B }{2}+\cot \frac{ A }{2}}=\tan \frac{ C }{2}=\frac{1}{\cot \frac{ C }{2}}$
$\Rightarrow\left(\cot \frac{ A }{2} \cot \frac{B}{2}-1\right) \cot \frac{ C }{2}=\cot \frac{ B }{2}+\cot \frac{ A }{2}$
$\Rightarrow \cot \frac{ A }{2} \cot \frac{B}{2} \cot \frac{ C }{2}=\cot \frac{ C }{2}+\cot \frac{ B }{2}+\cot \frac{ A }{2}$

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