If `x = 2 sin^(2) theta and y = 2 cos^(2) theta + 1` then x+ y is:

Options

A. 3
B. 2
C. 1
D. `1//2`

Answer

To solve the problem, we need to find the value of \( x + y \) given that \( x = 2 \sin^2 \theta \) and \( y = 2 \cos^2 \theta + 1 \).

### Step-by-step Solution:

1. **Write down the expressions for x and y:**
\[ x = 2 \sin^2 \theta \]
\[ y = 2 \cos^2 \theta + 1 \]

2. **Add the expressions for x and y:**
\[ x + y = 2 \sin^2 \theta + (2 \cos^2 \theta + 1) \]

3. **Combine like terms:**
\[ x + y = 2 \sin^2 \theta + 2 \cos^2 \theta + 1 \]

4. **Factor out the common term (2):**
\[ x + y = 2 (\sin^2 \theta + \cos^2 \theta) + 1 \]

5. **Use the Pythagorean identity:**
Recall that \( \sin^2 \theta + \cos^2 \theta = 1 \).
\[ x + y = 2(1) + 1 \]

6. **Simplify the expression:**
\[ x + y = 2 + 1 = 3 \]

### Final Answer:
Thus, the value of \( x + y \) is \( 3 \).