Here's a clear, educational, and well-structured solution to the given problem.

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Key Concepts Used:

1. Properties of Determinants:
   • The value of a determinant remains unchanged if we apply the operation $C_i \to C_i + k C_j$ (or $R_i \to R_i + k R_j$), where $C_i$ (or $R_i$) denotes the $i$-th column (or row). This property is crucial for simplifying determinants by creating zeros.
   • If any row or column of a determinant contains all zeros, the value of the determinant is zero.
2. Expansion of a Determinant: A $3 \times 3$ determinant can be expanded along any row or column. Expanding along a row or column that contains more zeros significantly reduces the number of terms to calculate, simplifying the process.
3. Trigonometric Identities: The fundamental identity $\sin^2 \theta + \cos^2 \theta = 1$ is frequently used to simplify trigonometric expressions.
4. Solving Trigonometric Equations: To solve an equation of the form $\sin \theta = k$, we find the principal values and then use the general solution $\theta = n\pi + (-1)^n \alpha$, where $\alpha$ is a principal value, or by identifying solutions in specific intervals.

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Step-by-Step Derivation:

We are asked to find the solutions of the equation: \left| {\matrix{ {1 + {{\sin }^2}x} & {{{\sin }^2}x} & {{{\sin }^2}x} \cr {{{\cos }^2}x} & {1 + {{\cos }^2}x} & {{{\cos }^2}x} \cr {4\sin 2x} & {4\sin 2x} & {1 + 4\sin 2x} \cr } } \right| = 0 \quad (0 < x < \pi )

Step 1: Simplify the Determinant using Column Operations

Our goal is to create as many zeros as possible in a row or column to simplify the expansion. Observe that $C_2$ and $C_3$ are very similar to $C_1$. Let's apply the column operations $C_2 \to C_2 - C_1$ and $C_3 \to C_3 - C_1$. This operation does not change the value of the determinant.

• For the first row:
  • $C_2$: $\sin^2 x - (1 + \sin^2 x) = -1$
  • $C_3$: $\sin^2 x - (1 + \sin^2 x) = -1$
• For the second row:
  • $C_2$: $(1 + \cos^2 x) - \cos^2 x = 1$
  • $C_3$: $\cos^2 x - \cos^2 x = 0$
• For the third row:
  • $C_2$: $4\sin 2x - 4\sin 2x = 0$
  • $C_3$: $(1 + 4\sin 2x) - 4\sin 2x = 1$

Applying these operations, the determinant simplifies to: \left| {\matrix{ {1 + {{\sin }^2}x} & {-1} & {-1} \cr {{{\cos }^2}x} & {1} & {0} \cr {4\sin 2x} & {0} & {1} \cr } } \right| = 0 This form is much easier to expand as it contains several zeros.

Step 2: Expand the Determinant

We will expand the determinant along the second row ($R_2$) because it contains a zero, which reduces one term in the expansion. Recall the expansion formula for a $3 \times 3$ determinant: \left| {\matrix{ a & b & c \cr d & e & f \cr g & h & i \cr } } \right| = d \cdot C_{21} + e \cdot C_{22} + f \cdot C_{23} = -d(bi-ch) + e(ai-cg) - f(ah-bg) where $C_{ij}$ is the cofactor of the element at row $i$, column $j$.

Expanding along $R_2$: -(\cos^2 x) \left| {\matrix{ -1 & -1 \cr 0 & 1 \cr } } \right| + (1) \left| {\matrix{ {1 + {{\sin }^2}x} & {-1} \cr {4\sin 2x} & {1} \cr } } \right| - (0) \left| {\matrix{ {1 + {{\sin }^2}x} & {-1} \cr {4\sin 2x} & {0} \cr } } \right| = 0 $-(\cos^2 x) ((-1)(1) - (-1)(0)) + (1) ((1 + \sin^2 x)(1) - (-1)(4\sin 2x)) - 0 = 0$ $-(\cos^2 x) (-1) + (1 + \sin^2 x + 4\sin 2x) = 0$ $\cos^2 x + 1 + \sin^2 x + 4\sin 2x = 0$ Using the trigonometric identity $\sin^2 x + \cos^2 x = 1$: $1 + 1 + 4\sin 2x = 0$ $2 + 4\sin 2x = 0$ $4\sin 2x = -2$ $\sin 2x = -\frac{1}{2}$

Step 3: Solve the Trigonometric Equation for the Given Interval

We need to solve $\sin 2x = -\frac{1}{2}$ for $0 < x < \pi$. First, let's determine the range for $2x$. Since $0 < x < \pi$, we have $0 < 2x < 2\pi$.

The sine function is negative in the third and fourth quadrants. The reference angle for which $\sin \theta = \frac{1}{2}$ is $\frac{\pi}{6}$. So, the values for $2x$ in the interval $(0, 2\pi)$ are:

1. Third quadrant solution: $2x = \pi + \frac{\pi}{6} = \frac{7\pi}{6}$
2. Fourth quadrant solution: $2x = 2\pi - \frac{\pi}{6} = \frac{11\pi}{6}$

Now, we solve for $x$:

1. From $2x = \frac{7\pi}{6}$, we get $x = \frac{7\pi}{12}$.
2. From $2x = \frac{11\pi}{6}$, we get $x = \frac{11\pi}{12}$.

Both $x = \frac{7\pi}{12}$ and $x = \frac{11\pi}{12}$ lie within the given interval $(0, \pi)$. ($\frac{7\pi}{12} \approx 0.58\pi$ and $\frac{11\pi}{12} \approx 0.91\pi$).

Thus, the solutions are $x = \frac{7\pi}{12}$ and $x = \frac{11\pi}{12}$.

Comparing these solutions with the given options: (A) ${\pi \over {12}},{\pi \over 6}$ (B) ${\pi \over 6},{{5\pi } \over 6}$ (C) ${{5\pi } \over {12}},{{7\pi } \over {12}}$ (D) ${{7\pi } \over {12}},{{11\pi } \over {12}}$

Our derived solutions match option (D).

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Tips for Solving Determinant Problems:

• Look for patterns: Always scan the determinant for identical rows/columns, common factors, or elements that could become zero with simple operations.
• Prioritize zeros: The most effective strategy is to use row/column operations to create as many zeros as possible in a single row or column. This drastically simplifies the determinant expansion.
• Use trigonometric identities: Be ready to apply fundamental identities like $\sin^2 x + \cos^2 x = 1$ to simplify expressions after expansion.
• Check interval for trigonometric equations: Always ensure your solutions for trigonometric equations fall within the specified domain.

Common Mistakes to Avoid:

• Sign errors during expansion: Be extremely careful with the alternating signs when expanding a determinant (e.g., for a $3 \times 3$ matrix, the pattern is $+ - +$ for the first row, $- + -$ for the second, etc.).
• Incorrect determinant operations: Ensure you apply row/column operations correctly. Remember that $C_i \to kC_i$ changes the determinant value by a factor of $k$, while $C_i \to C_i + kC_j$ does not.
• Missing solutions: For trigonometric equations, remember that there are often multiple solutions within a given interval, and general solutions repeat periodically. Always check all quadrants.
• Solutions outside the domain: After finding potential solutions, always verify that they lie within the specified interval (e.g., $0 < x < \pi$).

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Summary/Key Takeaway:

This problem demonstrates a standard approach to solving determinant equations involving trigonometric functions. The key steps involve:

1. Systematically simplifying the determinant using column/row operations to introduce zeros.
2. Expanding the simplified determinant.
3. Utilizing trigonometric identities to reduce the resulting equation.
4. Solving the trigonometric equation while carefully considering the given domain for the variable. This method efficiently transforms a complex determinant problem into a manageable trigonometric equation.

The solutions to the equation are $\boxed{{7\pi \over {12}},{{11\pi } \over {12}}}$.