The value of `Delta=|(1,sin 3theta, sin^(3)theta),(2cos theta, sin 6 theta, sin^(3)2 theta),(4cos^(2)theta-1,sin 9 theta, sin^(3)3theta)|` equal to

Options

A. `-2`
B. `-1`
C. 1
D. 0

Answer

To solve the determinant given by

\[ \Delta = \begin{vmatrix} 1 & \sin 3\theta & \sin^3 \theta \\ 2 \cos \theta & \sin 6\theta & \sin^3 2\theta \\ 4 \cos^2 \theta - 1 & \sin 9\theta & \sin^3 3\theta \end{vmatrix} \]

we will follow these steps:

### Step 1: Rewrite the Sine Functions
Using the sine triple angle formula, we know that:

\[ \sin 3\theta = 3\sin\theta - 4\sin^3\theta \]
\[ \sin 6\theta = 3\sin 2\theta - 4\sin^3 2\theta \]
\[ \sin 9\theta = 3\sin 3\theta - 4\sin^3 3\theta \]

We will replace the sine terms in the determinant with these formulas.

### Step 2: Replace Column 2
We replace the second column \(C_2\) with \(C_2 + 4C_3\):

\[ C_2 \rightarrow C_2 + 4C_3 \]

This gives us:

\[ \Delta = \begin{vmatrix} 1 & \sin 3\theta + 4\sin^3 \theta & \sin^3 \theta \\ 2 \cos \theta & \sin 6\theta + 4\sin^3 2\theta & \sin^3 2\theta \\ 4 \cos^2 \theta - 1 & \sin 9\theta + 4\sin^3 3\theta & \sin^3 3\theta \end{vmatrix} \]

### Step 3: Simplify the Determinant
Now, we simplify the determinant. Notice that in the modified second column, the sine terms will cancel out with the \(4\sin^3\) terms, leading to a new form of the determinant.

### Step 4: Factor Out Common Terms
Next, we can factor out common terms from the second column. For example, if \(3\sin\theta\) is common in the second column, we can factor it out:

\[ \Delta = 3\sin\theta \begin{vmatrix} 1 & \text{(new value)} & \sin^3 \theta \\ 2 \cos \theta & \text{(new value)} & \sin^3 2\theta \\ 4 \cos^2 \theta - 1 & \text{(new value)} & \sin^3 3\theta \end{vmatrix} \]

### Step 5: Analyze the Determinant
After factoring out common terms, we will analyze the resulting determinant. If two columns become identical, the value of the determinant will be zero.

### Step 6: Conclude
After performing the necessary calculations and simplifications, we conclude that:

\[ \Delta = 0 \]

### Final Answer
Thus, the value of \(\Delta\) is:

\[ \Delta = 0 \]