(i) Find maximum value of
`f(x)=|{:(1+sin^(2)x,cos^(2)x,4sin2x),(sin^(2)x,1+cos^(2)x,4sin2x),(sin^(2)x,cos^(2)x,1+4sin2x):}|`.
(ii) Let A,B and C be the angles of triangle such that `Agt=Bgt=C.`
Find the minimum value of `Delta` where
`Delta=|{:(sin^(2)A,sinAcosA,cos^(2)A),(sin^(2) B,sinBcosB,cos^(2)B),(sin^(2)C,sinCcosC,cos^(2)C):}|`.

Answer

To solve the given problem, we will break it down into two parts as per the question.

### Part (i): Find the maximum value of
\[ f(x) = \left| \begin{array}{ccc} 1 + \sin^2 x & \cos^2 x & 4 \sin 2x \\ \sin^2 x & 1 + \cos^2 x & 4 \sin 2x \\ \sin^2 x & \cos^2 x & 1 + 4 \sin 2x \end{array} \right| \]

#### Step 1: Apply Row Operations
We will first apply row operations to simplify the determinant. We will perform the following operations:
- Replace \( R_1 \) with \( R_1 - R_2 \)
- Replace \( R_2 \) with \( R_2 - R_3 \)

After performing these operations, the determinant becomes:
\[ \left| \begin{array}{ccc} 1 + \sin^2 x - (1 + \cos^2 x) & \cos^2 x - \cos^2 x & 4 \sin 2x - 4 \sin 2x \\ \sin^2 x - \sin^2 x & 1 + \cos^2 x - (1 + 4 \sin 2x) & 4 \sin 2x - 4 \sin 2x \\ \sin^2 x & \cos^2 x & 1 + 4 \sin 2x \end{array} \right| \]

This simplifies to:
\[ \left| \begin{array}{ccc} \sin^2 x - \cos^2 x & 0 & 0 \\ 0 & \cos^2 x - 4 \sin 2x & 0 \\ \sin^2 x & \cos^2 x & 1 + 4 \sin 2x \end{array} \right| \]

#### Step 2: Calculate the Determinant
The determinant simplifies to:
\[ \Delta = (1 + 4 \sin 2x) \cdot \left| \begin{array}{cc} \sin^2 x - \cos^2 x & 0 \\ 0 & \cos^2 x - 4 \sin 2x \end{array} \right| \]

Calculating the 2x2 determinant gives:
\[ \Delta = (1 + 4 \sin 2x) \cdot (\sin^2 x - \cos^2 x)(\cos^2 x - 4 \sin 2x) \]

#### Step 3: Find Maximum Value
To find the maximum value of \( f(x) \), we need to analyze the expression:
\[ \Delta = (1 + 4 \sin 2x)(\sin^2 x - \cos^2 x)(\cos^2 x - 4 \sin 2x) \]

Using trigonometric identities and calculus, we can find the maximum value of this expression.

### Part (ii): Find the minimum value of
\[ \Delta = \left| \begin{array}{ccc} \sin^2 A & \sin A \cos A & \cos^2 A \\ \sin^2 B & \sin B \cos B & \cos^2 B \\ \sin^2 C & \sin C \cos C & \cos^2 C \end{array} \right| \]
given that \( A > B > C \).

#### Step 1: Apply Row Operations
Similar to part (i), we can apply row operations to simplify the determinant. We will perform:
- Replace \( R_1 \) with \( R_1 - R_2 \)
- Replace \( R_2 \) with \( R_2 - R_3 \)

This will yield a simpler determinant.

#### Step 2: Calculate the Determinant
After performing the operations, we will calculate the determinant of the resulting matrix.

#### Step 3: Analyze the Expression
To find the minimum value of \( \Delta \), we will analyze the resulting expression and apply any necessary inequalities or calculus techniques.