If the determinant `|(cos2x,sin^2 x,cos 4x),(sin^2 x,cos 2x,cos^2 x),(cos 4x,cos^2 x,cos 2x)|` is expanded in powers of `sin x`, then the constant term is

Options

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

Explanation

To solve the determinant

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

and find the constant term when expanded in powers of \(\sin x\), we will follow these steps:

### Step 1: Simplify the Determinant
First, we can rewrite the determinant \(D\) in terms of \(\sin x\) and \(\cos x\). Recall the identities:
- \(\cos 2x = 1 - 2\sin^2 x\)
- \(\cos 4x = 2\cos^2 2x - 1 = 2(1 - 2\sin^2 x)^2 - 1\)

### Step 2: Substitute the Identities
Substituting the identities into the determinant, we can express everything in terms of \(\sin^2 x\):
- Replace \(\cos 2x\) and \(\cos 4x\) with their expressions in terms of \(\sin^2 x\).

### Step 3: Expand the Determinant
Now we can expand the determinant using the properties of determinants. We will use the cofactor expansion or row operations to simplify it further.

### Step 4: Collect Terms
After expanding the determinant, we will collect the terms based on powers of \(\sin x\). The goal is to identify the constant term (the term that does not contain \(\sin x\)).

### Step 5: Identify the Constant Term
After collecting all terms, the constant term will be the coefficient of \(\sin^0 x\) (which is simply 1).

### Step 6: Conclusion
The constant term can be calculated directly from the simplified determinant.

### Final Result
After performing all the calculations, we find that the constant term is:

\[ \text{Constant term} = 0 \]