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  4. In any triangle ABC, which is not right angled ∑ cos A ⋅ textcosecB ⋅ textcosec C is equal to

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Q. In any triangle $ABC$, which is not right angled $\sum \cos A \cdot \text{cosecB} \cdot \text{cosec} C$ is equal to

Trigonometric Functions

Solution:

$\sum \cos A \text{cosec} B \text{cosec} C=\frac{\cos A}{\sin B \sin C}+\frac{\cos B}{\sin A \sin C}+\frac{\cos C}{\sin A \sin B}$
$=\frac{\cos A \sin A+\cos B \sin B+\cos C \sin C}{\sin A \sin B \sin C}$
$=\frac{\sin 2 A+\sin 2 B+\sin 2 C}{2 \sin A \sin B \sin C}=\frac{4 \sin A \sin B \sin C}{2 \sin A \sin B \sin C}=2$
(using conditional identity)