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. y
C. 3y
D. none of these

Explanation

To solve the problem, we start with the given equations:

1. \( x = \sin A + \sin 2A \)
2. \( y = \cos A + \cos 2A \)

We need to find the value of \( (x^2 + y^2)(x^2 + y^2 - 3) \).

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

First, we square both expressions:

\[ x^2 = (\sin A + \sin 2A)^2 = \sin^2 A + \sin^2 2A + 2 \sin A \sin 2A \]

\[ y^2 = (\cos A + \cos 2A)^2 = \cos^2 A + \cos^2 2A + 2 \cos A \cos 2A \]

### Step 2: Add \( x^2 \) and \( y^2 \)

Now, we add \( x^2 \) and \( y^2 \):

\[ x^2 + y^2 = (\sin^2 A + \sin^2 2A + 2 \sin A \sin 2A) + (\cos^2 A + \cos^2 2A + 2 \cos A \cos 2A) \]

Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \):

\[ x^2 + y^2 = (\sin^2 A + \cos^2 A) + (\sin^2 2A + \cos^2 2A) + 2(\sin A \sin 2A + \cos A \cos 2A) \]

This simplifies to:

\[ x^2 + y^2 = 1 + 1 + 2(\sin A \sin 2A + \cos A \cos 2A) \]

### Step 3: Use the Cosine Addition Formula

The expression \( \sin A \sin 2A + \cos A \cos 2A \) can be rewritten using the cosine addition formula:

\[ \sin A \sin 2A + \cos A \cos 2A = \cos(2A - A) = \cos A \]

Thus, we have:

\[ x^2 + y^2 = 2 + 2 \cos A \]

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

Now, we substitute \( x^2 + y^2 \) into the expression we need to evaluate:

Let \( z = x^2 + y^2 \), then:

\[ z = 2 + 2 \cos A \]

We need to find:

\[ (z)(z - 3) = z(z - 3) = z^2 - 3z \]

### Step 5: Calculate \( z^2 \) and \( 3z \)

First, calculate \( z^2 \):

\[ z^2 = (2 + 2 \cos A)^2 = 4(1 + \cos A)^2 = 4(1 + 2 \cos A + \cos^2 A) \]

Now, calculate \( 3z \):

\[ 3z = 3(2 + 2 \cos A) = 6 + 6 \cos A \]

### Step 6: Final Expression

Now, we substitute back:

\[ z^2 - 3z = 4(1 + 2 \cos A + \cos^2 A) - (6 + 6 \cos A) \]

Simplifying gives:

\[ = 4 + 8 \cos A + 4 \cos^2 A - 6 - 6 \cos A \]
\[ = -2 + 2 \cos A + 4 \cos^2 A \]

### Step 7: Factor and Simplify

This can be factored as:

\[ = 2(2 \cos^2 A + \cos A - 1) \]

### Conclusion

Thus, the value of \( (x^2 + y^2)(x^2 + y^2 - 3) \) simplifies to:

\[ = 2y \]

The correct answer is:

\[ \boxed{2y} \]