If `x=sintheta+costhetaandy=sintheta.costheta`, then what is the value of `x^(4)-4x^(2)y-2x^(2)+4y^(2)+4y+1`?

Options

A. 0
B. 1
C. 2
D. None of these

Explanation

To solve the expression \( x^4 - 4x^2y - 2x^2 + 4y^2 + 4y + 1 \) given that \( x = \sin \theta + \cos \theta \) and \( y = \sin \theta \cos \theta \), we will substitute the values of \( x \) and \( y \) into the expression step by step.

### Step 1: Calculate \( x^2 \)
First, we need to calculate \( x^2 \):
\[ x^2 = (\sin \theta + \cos \theta)^2 = \sin^2 \theta + \cos^2 \theta + 2\sin \theta \cos \theta \]
Using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \), we get:
\[ x^2 = 1 + 2y \]

### Step 2: Calculate \( x^4 \)
Next, we calculate \( x^4 \):
\[ x^4 = (x^2)^2 = (1 + 2y)^2 = 1 + 4y + 4y^2 \]

### Step 3: Calculate \( 4x^2y \)
Now, we calculate \( 4x^2y \):
\[ 4x^2y = 4(1 + 2y)y = 4y + 8y^2 \]

### Step 4: Calculate \( 2x^2 \)
Next, we calculate \( 2x^2 \):
\[ 2x^2 = 2(1 + 2y) = 2 + 4y \]

### Step 5: Substitute into the expression
Now we substitute \( x^4 \), \( 4x^2y \), and \( 2x^2 \) into the original expression:
\[ x^4 - 4x^2y - 2x^2 + 4y^2 + 4y + 1 \]
Substituting the values we calculated:
\[ (1 + 4y + 4y^2) - (4y + 8y^2) - (2 + 4y) + 4y^2 + 4y + 1 \]

### Step 6: Simplify the expression
Now we simplify the expression:
\[ = 1 + 4y + 4y^2 - 4y - 8y^2 - 2 - 4y + 4y^2 + 4y + 1 \]
Combining like terms:
- Constant terms: \( 1 - 2 + 1 = 0 \)
- \( y \) terms: \( 4y - 4y - 4y + 4y = 0 \)
- \( y^2 \) terms: \( 4y^2 - 8y^2 + 4y^2 = 0 \)

### Final Result
Thus, the entire expression simplifies to:
\[ 0 \]