If `sin A + sin 2A =x` and `cos A + cos 2A = y`, then `( x^(2) + y^(2) ) ( x^(2) + y ^(2) -3)=`

Options

A. 2y
B. 2x
C. `x+y`
D. none

Answer

To solve the problem, we need to find the expression \((x^2 + y^2)(x^2 + y^2 - 3)\) given that \(x = \sin A + \sin 2A\) and \(y = \cos A + \cos 2A\).

### Step 1: Calculate \(x^2\) and \(y^2\)

1. **Calculate \(x^2\)**:
\[ x = \sin A + \sin 2A \]
Using the double angle formula, \(\sin 2A = 2 \sin A \cos A\):
\[ x = \sin A + 2 \sin A \cos A = \sin A (1 + 2 \cos A) \]
Therefore,
\[ x^2 = (\sin A (1 + 2 \cos A))^2 = \sin^2 A (1 + 2 \cos A)^2 \]

2. **Calculate \(y^2\)**:
\[ y = \cos A + \cos 2A \]
Using the double angle formula, \(\cos 2A = 2 \cos^2 A - 1\):
\[ y = \cos A + (2 \cos^2 A - 1) = 2 \cos^2 A + \cos A - 1 \]
Therefore,
\[ y^2 = (2 \cos^2 A + \cos A - 1)^2 \]

### Step 2: Calculate \(x^2 + y^2\)

Now we need to compute \(x^2 + y^2\):
\[ x^2 + y^2 = \sin^2 A (1 + 2 \cos A)^2 + (2 \cos^2 A + \cos A - 1)^2 \]

### Step 3: Simplify \(x^2 + y^2\)

1. **Expand \(x^2\)**:
\[ x^2 = \sin^2 A (1 + 4 \cos A + 4 \cos^2 A) = \sin^2 A + 4 \sin^2 A \cos A + 4 \sin^2 A \cos^2 A \]

2. **Expand \(y^2\)**:
\[ y^2 = (2 \cos^2 A + \cos A - 1)^2 = 4 \cos^4 A + 4 \cos^3 A - 4 \cos^2 A + \cos^2 A - 2 \cos A + 1 \]
Combine like terms.

### Step 4: Combine \(x^2 + y^2\)

Combine the expanded forms of \(x^2\) and \(y^2\) to get \(x^2 + y^2\).

### Step 5: Calculate \((x^2 + y^2)(x^2 + y^2 - 3)\)

Finally, we need to compute:
\[ (x^2 + y^2)(x^2 + y^2 - 3) \]

### Final Answer

After performing all calculations and simplifications, we will arrive at the final expression.