If the determinant `|(cos2theta, sin^2 theta, cos4theta),(sin^2 theta,cos 2theta,cos^2theta),(cos4theta,cos^2theta,cos2theta)|` is expanded in powers of `sintheta` , then the constant term in the expansion is

Options

A. `-1`
B. 1
C. 2
D. none

Answer

To find the constant term in the expansion of the determinant

\[ D = \begin{vmatrix} \cos 2\theta & \sin^2 \theta & \cos 4\theta \\ \sin^2 \theta & \cos 2\theta & \cos^2 \theta \\ \cos 4\theta & \cos^2 \theta & \cos 2\theta \end{vmatrix} \]

in powers of \(\sin \theta\), we will evaluate the determinant at \(\theta = 0\) since the constant term corresponds to the value of the determinant when \(\sin \theta = 0\).

### Step 1: Evaluate the trigonometric functions at \(\theta = 0\)

At \(\theta = 0\):
- \(\cos 2\theta = \cos 0 = 1\)
- \(\sin^2 \theta = \sin^2 0 = 0\)
- \(\cos 4\theta = \cos 0 = 1\)
- \(\cos^2 \theta = \cos^2 0 = 1\)

### Step 2: Substitute the values into the determinant

Substituting these values into the determinant gives:

\[ D = \begin{vmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 1 & 1 \end{vmatrix} \]

### Step 3: Calculate the determinant

Now we will calculate this determinant using the formula for a \(3 \times 3\) determinant:

\[ D = a(ei - fh) - b(di - fg) + c(dh - eg) \]

For our matrix:

- \(a = 1\), \(b = 0\), \(c = 1\)
- \(d = 0\), \(e = 1\), \(f = 1\)
- \(g = 1\), \(h = 1\), \(i = 1\)

Calculating each term:

1. \(ei - fh = 1 \cdot 1 - 1 \cdot 1 = 1 - 1 = 0\)
2. \(di - fg = 0 \cdot 1 - 1 \cdot 1 = 0 - 1 = -1\)
3. \(dh - eg = 0 \cdot 1 - 1 \cdot 1 = 0 - 1 = -1\)

Now substituting these back into the determinant formula:

\[ D = 1 \cdot 0 - 0 \cdot (-1) + 1 \cdot (-1) = 0 + 0 - 1 = -1 \]

### Conclusion

Thus, the constant term in the expansion of the determinant in powers of \(\sin \theta\) is

\[ \boxed{-1} \]