In `DeltaABC`, if `(sin A)/(c sin B) + (sin B)/(c) + (sin C)/(b) = (c)/(ab) + (b)/(ac) + (a)/(bc)`, then the value of angle A is

Options

A. `120^(@)`
B. `90^(@)`
C. `60^(@)`
D. `30^(@)`

Explanation

To solve the given equation in triangle \( \Delta ABC \):

\[ \frac{\sin A}{c \sin B} + \frac{\sin B}{c} + \frac{\sin C}{b} = \frac{c}{ab} + \frac{b}{ac} + \frac{a}{bc} \]

we will follow these steps:

### Step 1: Rewrite the equation using the sine rule
Using the sine rule, we know that:
- \( a = 2R \sin A \)
- \( b = 2R \sin B \)
- \( c = 2R \sin C \)

Substituting these into the equation gives:

\[ \frac{\frac{a}{2R}}{c \cdot \frac{b}{2R}} + \frac{\frac{b}{2R}}{c} + \frac{\frac{c}{2R}}{b} = \frac{c}{ab} + \frac{b}{ac} + \frac{a}{bc} \]

### Step 2: Simplify the left-hand side
The left-hand side simplifies to:

\[ \frac{a}{2R \cdot c \cdot \frac{b}{2R}} + \frac{b}{2R \cdot c} + \frac{c}{2R \cdot b} \]

This simplifies further to:

\[ \frac{a}{bc} + \frac{b}{2Rc} + \frac{c}{2Rb} \]

### Step 3: Simplify the right-hand side
The right-hand side is:

\[ \frac{c}{ab} + \frac{b}{ac} + \frac{a}{bc} \]

### Step 4: Equate both sides
Now we have:

\[ \frac{a}{bc} + \frac{b}{2Rc} + \frac{c}{2Rb} = \frac{c}{ab} + \frac{b}{ac} + \frac{a}{bc} \]

### Step 5: Cancel common terms
We can cancel \( \frac{a}{bc} \) from both sides:

\[ \frac{b}{2Rc} + \frac{c}{2Rb} = \frac{c}{ab} + \frac{b}{ac} \]

### Step 6: Cross-multiply and simplify
Cross-multiplying gives us:

\[ b^2 + c^2 = 2R(c + b) \]

### Step 7: Use the Law of Cosines
From the Law of Cosines, we know:

\[ a^2 = b^2 + c^2 - 2bc \cos A \]

### Step 8: Substitute and solve for \( A \)
Substituting \( b^2 + c^2 \) into the equation gives:

\[ a^2 = 2R(c + b) - 2bc \cos A \]

### Step 9: Solve for \( \cos A \)
Using the identity \( \sin^2 A + \cos^2 A = 1 \), we can find \( A \):

\[ \cos A = 0 \implies A = 90^\circ \]

### Conclusion
Thus, the value of angle \( A \) is:

\[ \boxed{90^\circ} \]