If A + B + C = pi , then frac{{cos A}}{{sin B | vedclass

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(C) Given $A + B + C = \pi$. Let $S = \frac{\cos A}{\sin B \sin C} + \frac{\cos B}{\sin C \sin A} + \frac{\cos C}{\sin A \sin B}$. Taking the common d

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$2$

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(C) Given $A + B + C = \pi$. Let $S = \frac{\cos A}{\sin B \sin C} + \frac{\cos B}{\sin C \sin A} + \frac{\cos C}{\sin A \sin B}$. Taking the common denominator $\sin A \sin B \sin C$,we get: $S = \frac{\cos A \sin A + \cos B \sin B + \cos C \sin C}{\sin A \sin B \sin C}$. Multiply numerator and denominator by $2$: $S = \frac{2 \sin A \cos A + 2 \sin B \cos B + 2 \sin C \cos C}{2 \sin A \sin B \sin C} = \frac{\sin 2A + \sin 2B + \sin 2C}{2 \sin A \sin B \sin C}$. Using the identity for a triangle where $A + B + C = \pi$,$\sin 2A + \sin 2B + \sin 2C = 4 \sin A \sin B \sin C$. Substituting this into the expression: $S = \frac{4 \sin A \sin B \sin C}{2 \sin A \sin B \sin C} = 2$.

If $A + B + C = \pi ,$ then $\frac{{\cos A}}{{\sin B\sin C}} + \frac{{\cos B}}{{\sin C\sin A}} + \frac{{\cos C}}{{\sin A\sin B}} = $

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