If `3 sin theta + 5 cos theta = 5`, then the value of `5 sin theta - 3 cos theta` will be.

Options

A. `(a) +3`
B. (b)`+5`
C. (c)`+2`
D. (d) `+1`

Explanation

To solve the problem, we need to find the value of \( 5 \sin \theta - 3 \cos \theta \) given that \( 3 \sin \theta + 5 \cos \theta = 5 \).

### Step-by-Step Solution:

1. **Identify the equations:**
We have two expressions:
- Equation 1: \( 3 \sin \theta + 5 \cos \theta = 5 \)
- Equation 2: \( x = 5 \sin \theta - 3 \cos \theta \) (This is what we need to find.)

2. **Square both equations:**
We will square both equations to help eliminate the sine and cosine terms later.
- Squaring Equation 1:
\[ (3 \sin \theta + 5 \cos \theta)^2 = 5^2 \]
Expanding this gives:
\[ 9 \sin^2 \theta + 30 \sin \theta \cos \theta + 25 \cos^2 \theta = 25 \]

- Squaring Equation 2:
\[ (5 \sin \theta - 3 \cos \theta)^2 = x^2 \]
Expanding this gives:
\[ 25 \sin^2 \theta - 30 \sin \theta \cos \theta + 9 \cos^2 \theta = x^2 \]

3. **Combine the squared equations:**
Now we will add the two squared equations:
\[ (9 \sin^2 \theta + 30 \sin \theta \cos \theta + 25 \cos^2 \theta) + (25 \sin^2 \theta - 30 \sin \theta \cos \theta + 9 \cos^2 \theta) = 25 + x^2 \]
Simplifying the left side:
\[ (9 + 25) \sin^2 \theta + (25 + 9) \cos^2 \theta + (30 - 30) \sin \theta \cos \theta = 25 + x^2 \]
This simplifies to:
\[ 34 \sin^2 \theta + 34 \cos^2 \theta = 25 + x^2 \]

4. **Use the Pythagorean identity:**
We know that \( \sin^2 \theta + \cos^2 \theta = 1 \):
\[ 34(1) = 25 + x^2 \]
Thus:
\[ 34 = 25 + x^2 \]

5. **Solve for \( x^2 \):**
Rearranging gives:
\[ x^2 = 34 - 25 \]
Therefore:
\[ x^2 = 9 \]

6. **Find the value of \( x \):**
Taking the square root gives:
\[ x = \pm 3 \]
Hence, the value of \( 5 \sin \theta - 3 \cos \theta \) can be either \( 3 \) or \( -3 \).

### Conclusion:
The value of \( 5 \sin \theta - 3 \cos \theta \) is \( \pm 3 \).