If `x=asinthetaandy=acostheta` then find the value of `x^(2)+y^(2)`

Answer

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

### Step-by-Step Solution:

1. **Write down the expressions for \( x \) and \( y \)**:
\[ x = a \sin \theta \]
\[ y = a \cos \theta \]

2. **Square both \( x \) and \( y \)**:
\[ x^2 = (a \sin \theta)^2 = a^2 \sin^2 \theta \]
\[ y^2 = (a \cos \theta)^2 = a^2 \cos^2 \theta \]

3. **Add \( x^2 \) and \( y^2 \)**:
\[ x^2 + y^2 = a^2 \sin^2 \theta + a^2 \cos^2 \theta \]

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

5. **Use the Pythagorean identity**:
We know that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Therefore:
\[ x^2 + y^2 = a^2 \cdot 1 \]

6. **Final result**:
\[ x^2 + y^2 = a^2 \]

### Final Answer:
The value of \( x^2 + y^2 \) is \( a^2 \).