If A + B + C =180^ , then the value of ( B + C )( C + A )( A + B)… - Vidyadip

MCQ

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

A
sec A sec B sec C
✓
cosec A cosec B cosec C
C
tan A tan B tan C
D
1

✓

Answer

Correct option: B.

cosec A cosec B cosec C

(B)
$\cot B +\cot C =\frac{\sin C \cos B +\sin B \cos C }{\sin B \sin C }$
$=\frac{\sin (B+C)}{\sin B \sin C}$
$=\frac{\sin \left(180^{\circ}- A \right)}{\sin B \sin C }$
$=\frac{\sin A }{\sin B \sin C }$
Similarly, $\cot C +\cot A =\frac{\sin B }{\sin C \sin A }$and $\cot A +\cot B =\frac{\sin C }{\sin A \sin B }$
$\therefore(\cot B +\cot C )(\cot C +\cot A )(\cot A +\cot B )$
$=\frac{\sin A}{\sin B \sin C} \cdot \frac{\sin B}{\sin C \sin A} \cdot \frac{\sin C}{\sin A \sin B}$
$=\operatorname{cosec} A \operatorname{cosec} B \operatorname{cosec} C$

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