In a triangle `sin^(4)A + sin^(4)B + sin^(4)C = sin^(2)B sin^(2)C + 2sin^(2) C sin^(2)A + 2sin^(2)A sin^(2)B`, then its angle A is equal to-

Options

A. `(pi)/6,(5pi)/6`
B. `(pi)/3,(5pi)/6`
C. `(5pi)/6,(2pi)/3`
D. none of these

Answer

To solve the equation \( \sin^4 A + \sin^4 B + \sin^4 C = \sin^2 B \sin^2 C + 2 \sin^2 C \sin^2 A + 2 \sin^2 A \sin^2 B \) and find the angle \( A \), we can follow these steps:

### Step 1: Rewrite the equation using the sine rule
Using the sine rule, we know that \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \). We can express \( \sin A, \sin B, \sin C \) in terms of the sides \( a, b, c \):
\[ \sin A = \frac{a}{k}, \quad \sin B = \frac{b}{k}, \quad \sin C = \frac{c}{k} \]
where \( k \) is a constant.

### Step 2: Substitute the sine values into the equation
Substituting these values into the equation gives:
\[ \left(\frac{a}{k}\right)^4 + \left(\frac{b}{k}\right)^4 + \left(\frac{c}{k}\right)^4 = \left(\frac{b}{k}\right)^2 \left(\frac{c}{k}\right)^2 + 2 \left(\frac{c}{k}\right)^2 \left(\frac{a}{k}\right)^2 + 2 \left(\frac{a}{k}\right)^2 \left(\frac{b}{k}\right)^2 \]
Multiplying through by \( k^4 \) (to eliminate the denominators) results in:
\[ a^4 + b^4 + c^4 = b^2 c^2 + 2c^2 a^2 + 2a^2 b^2 \]

### Step 3: Rearranging the equation
Rearranging gives:
\[ a^4 + b^4 + c^4 - 2a^2b^2 - 2a^2c^2 - 2b^2c^2 = 0 \]

### Step 4: Recognizing a perfect square
We can recognize that the left-hand side can be factored as:
\[ (b^2 + c^2 - a^2)^2 = 3b^2c^2 \]

### Step 5: Using the cosine rule
From the cosine rule, we know:
\[ \cos A = \frac{b^2 + c^2 - a^2}{2bc} \]
Substituting this into our equation gives:
\[ (b^2 + c^2 - a^2)^2 = 3b^2c^2 \]
This leads to:
\[ (2bc \cos A)^2 = 3b^2c^2 \]

### Step 6: Simplifying the equation
Dividing both sides by \( b^2c^2 \) gives:
\[ 4 \cos^2 A = 3 \]
Thus,
\[ \cos^2 A = \frac{3}{4} \]

### Step 7: Finding angle A
Taking the square root gives:
\[ \cos A = \pm \frac{\sqrt{3}}{2} \]
This implies:
\[ A = \frac{\pi}{6} \quad \text{or} \quad A = \frac{5\pi}{6} \]

### Conclusion
Thus, the angle \( A \) can be either \( \frac{\pi}{6} \) or \( \frac{5\pi}{6} \).