If `sintheta-costheta=(7)/(13)and0ltthetalt90^(@)` , then the value of `sintheta+costheta` is

Options

A. `(17)/(13)`
B. `(13)/(17)`
C. `(1)/(13)`
D. `(1)/(17)`

Answer

To solve the problem where \( \sin \theta - \cos \theta = \frac{7}{13} \) and \( 0 < \theta < 90^\circ \), we want to find the value of \( \sin \theta + \cos \theta \).

### Step-by-Step Solution:

1. **Let \( \sin \theta + \cos \theta = x \)**:
\[ \sin \theta + \cos \theta = x \]

2. **Square both equations**:
\[ (\sin \theta - \cos \theta)^2 = \left(\frac{7}{13}\right)^2 \]
\[ (\sin \theta + \cos \theta)^2 = x^2 \]

3. **Expand both squared equations**:
\[ \sin^2 \theta - 2 \sin \theta \cos \theta + \cos^2 \theta = \frac{49}{169} \]
\[ \sin^2 \theta + 2 \sin \theta \cos \theta + \cos^2 \theta = x^2 \]

4. **Use the Pythagorean identity**:
Recall that \( \sin^2 \theta + \cos^2 \theta = 1 \):
\[ 1 - 2 \sin \theta \cos \theta = \frac{49}{169} \]
\[ 1 + 2 \sin \theta \cos \theta = x^2 \]

5. **From the first equation, isolate \( 2 \sin \theta \cos \theta \)**:
\[ -2 \sin \theta \cos \theta = \frac{49}{169} - 1 \]
\[ -2 \sin \theta \cos \theta = \frac{49}{169} - \frac{169}{169} = \frac{-120}{169} \]
\[ 2 \sin \theta \cos \theta = \frac{120}{169} \]

6. **Substitute \( 2 \sin \theta \cos \theta \) into the second equation**:
\[ 1 + \frac{120}{169} = x^2 \]
\[ x^2 = \frac{169}{169} + \frac{120}{169} = \frac{289}{169} \]

7. **Take the square root**:
\[ x = \sqrt{\frac{289}{169}} = \frac{17}{13} \]

### Final Answer:
Thus, the value of \( \sin \theta + \cos \theta \) is \( \frac{17}{13} \).

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