In a triangle ABC the expression `acosBcosC+bcosC cosA+c cosA cosB` equals to :

Options

A. `(rs)/(R )`
B. `(r )/(sR)`
C. `(R )/(rs)`
D. `(Rs)/(r )`

Answer

To solve the expression \( A \cos B \cos C + B \cos C \cos A + C \cos A \cos B \) in triangle \( ABC \), we will follow these steps:

### Step 1: Rewrite the Expression
We start with the expression:
\[ E = A \cos B \cos C + B \cos C \cos A + C \cos A \cos B \]

### Step 2: Factor Out Common Terms
We can factor out \( \cos A \cos B \cos C \) from the expression:
\[ E = \cos A \cos B \cos C \left( \frac{A}{\cos A} + \frac{B}{\cos B} + \frac{C}{\cos C} \right) \]

### Step 3: Use the Sine Rule
According to the sine rule, we have:
\[ \frac{A}{\sin A} = \frac{B}{\sin B} = \frac{C}{\sin C} = 2R \]
where \( R \) is the circumradius of triangle \( ABC \). Thus, we can express \( A, B, C \) in terms of \( R \) and sine:
\[ A = 2R \sin A, \quad B = 2R \sin B, \quad C = 2R \sin C \]

### Step 4: Substitute into the Expression
Substituting these into our expression gives:
\[ E = \cos A \cos B \cos C \left( 2R \frac{\sin A}{\cos A} + 2R \frac{\sin B}{\cos B} + 2R \frac{\sin C}{\cos C} \right) \]
This simplifies to:
\[ E = 2R \cos A \cos B \cos C \left( \tan A + \tan B + \tan C \right) \]

### Step 5: Use the Identity for Tangents
Using the identity:
\[ \tan A + \tan B + \tan C = \tan A \tan B \tan C \]
we can rewrite the expression as:
\[ E = 2R \cos A \cos B \cos C \cdot \tan A \tan B \tan C \]

### Step 6: Substitute Tangent Values
Substituting the tangent values:
\[ \tan A = \frac{\sin A}{\cos A}, \quad \tan B = \frac{\sin B}{\cos B}, \quad \tan C = \frac{\sin C}{\cos C} \]
we can simplify further:
\[ E = 2R \cdot \frac{\sin A \sin B \sin C}{\cos A \cos B \cos C} \]

### Step 7: Final Expression
Using the sine rule again, we can express \( \sin A, \sin B, \sin C \) in terms of \( A, B, C \):
\[ E = 2R \cdot \frac{A}{2R} \cdot \frac{B}{2R} \cdot \frac{C}{2R} = \frac{ABC}{4R} \]

### Conclusion
Thus, the final value of the expression \( A \cos B \cos C + B \cos C \cos A + C \cos A \cos B \) is:
\[ \frac{ABC}{4R} \]