If x = 3sin`theta` + 4cos `theta` and y = 3cos `theta - `4 sin `theta` then prove that `x^(2) + y^(2) = 25`.

Answer

To prove that \( x^2 + y^2 = 25 \) given \( x = 3\sin\theta + 4\cos\theta \) and \( y = 3\cos\theta - 4\sin\theta \), we will follow these steps:

### Step 1: Write down the expressions for \( x \) and \( y \)
Given:
\[ x = 3\sin\theta + 4\cos\theta \]
\[ y = 3\cos\theta - 4\sin\theta \]

### Step 2: Calculate \( x^2 \) and \( y^2 \)
Now, we will calculate \( x^2 \) and \( y^2 \):
\[ x^2 = (3\sin\theta + 4\cos\theta)^2 = 9\sin^2\theta + 24\sin\theta\cos\theta + 16\cos^2\theta \]
\[ y^2 = (3\cos\theta - 4\sin\theta)^2 = 9\cos^2\theta - 24\sin\theta\cos\theta + 16\sin^2\theta \]

### Step 3: Add \( x^2 \) and \( y^2 \)
Now, we will add \( x^2 \) and \( y^2 \):
\[ x^2 + y^2 = (9\sin^2\theta + 24\sin\theta\cos\theta + 16\cos^2\theta) + (9\cos^2\theta - 24\sin\theta\cos\theta + 16\sin^2\theta) \]

### Step 4: Simplify the expression
Combining like terms:
\[ x^2 + y^2 = (9\sin^2\theta + 16\sin^2\theta) + (16\cos^2\theta + 9\cos^2\theta) + (24\sin\theta\cos\theta - 24\sin\theta\cos\theta) \]
\[ = 25\sin^2\theta + 25\cos^2\theta \]

### Step 5: Use the Pythagorean identity
We know that \( \sin^2\theta + \cos^2\theta = 1 \):
\[ x^2 + y^2 = 25(\sin^2\theta + \cos^2\theta) = 25 \cdot 1 = 25 \]

### Conclusion
Thus, we have proved that:
\[ x^2 + y^2 = 25 \]