If `A+B+C=pi`, then
`(cosA)/(sinBsinC)+(cosB)/(sinCsinA)+(cosC)/(sinAsinB)=?`

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the problem, we need to evaluate the expression:

\[ \frac{\cos A}{\sin B \sin C} + \frac{\cos B}{\sin C \sin A} + \frac{\cos C}{\sin A \sin B} \]

given that \( A + B + C = \pi \).

### Step 1: Choose values for A, B, and C
Since \( A + B + C = \pi \), we can choose \( A = B = C = \frac{\pi}{3} \) (which is 60 degrees). This is a convenient choice because it simplifies the trigonometric functions.

### Step 2: Calculate the trigonometric values
Using the values \( A = B = C = \frac{\pi}{3} \):
- \( \cos A = \cos \frac{\pi}{3} = \frac{1}{2} \)
- \( \sin A = \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \)

### Step 3: Substitute values into the expression
Now substituting these values into the expression:

\[ \frac{\cos A}{\sin B \sin C} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}} = \frac{\frac{1}{2}}{\frac{3}{4}} = \frac{1}{2} \cdot \frac{4}{3} = \frac{2}{3} \]

Since \( A, B, C \) are all equal, the other two terms will yield the same result:

\[ \frac{\cos B}{\sin C \sin A} = \frac{2}{3} \]

\[ \frac{\cos C}{\sin A \sin B} = \frac{2}{3} \]

### Step 4: Add the three terms together
Now we can add the three terms:

\[ \frac{2}{3} + \frac{2}{3} + \frac{2}{3} = 3 \cdot \frac{2}{3} = 2 \]

### Final Answer
Thus, the value of the expression is:

\[ \boxed{2} \]