If cot ` theta = (3)/(4)` find the value of :
`( sin theta- cos theta)/(sin theta+cos theta)`

Answer

To solve the problem, we need to find the value of \((\sin \theta - \cos \theta) / (\sin \theta + \cos \theta)\) given that \(\cot \theta = \frac{3}{4}\).

### Step-by-Step Solution:

1. **Understanding Cotangent**:
We know that \(\cot \theta = \frac{\cos \theta}{\sin \theta}\). Given \(\cot \theta = \frac{3}{4}\), we can express this in terms of sine and cosine:
\[ \frac{\cos \theta}{\sin \theta} = \frac{3}{4} \]
This implies that \(\cos \theta = 3k\) and \(\sin \theta = 4k\) for some positive value \(k\).

2. **Finding \(k\)**:
To find \(k\), we use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\):
\[ (4k)^2 + (3k)^2 = 1 \]
\[ 16k^2 + 9k^2 = 1 \]
\[ 25k^2 = 1 \]
\[ k^2 = \frac{1}{25} \]
\[ k = \frac{1}{5} \]

3. **Finding \(\sin \theta\) and \(\cos \theta\)**:
Now substitute \(k\) back to find \(\sin \theta\) and \(\cos \theta\):
\[ \sin \theta = 4k = 4 \times \frac{1}{5} = \frac{4}{5} \]
\[ \cos \theta = 3k = 3 \times \frac{1}{5} = \frac{3}{5} \]

4. **Substituting into the Expression**:
Now we substitute \(\sin \theta\) and \(\cos \theta\) into the expression \((\sin \theta - \cos \theta) / (\sin \theta + \cos \theta)\):
\[ \frac{\sin \theta - \cos \theta}{\sin \theta + \cos \theta} = \frac{\frac{4}{5} - \frac{3}{5}}{\frac{4}{5} + \frac{3}{5}} \]

5. **Simplifying the Expression**:
Simplifying the numerator and denominator:
\[ = \frac{\frac{4 - 3}{5}}{\frac{4 + 3}{5}} = \frac{\frac{1}{5}}{\frac{7}{5}} = \frac{1}{5} \times \frac{5}{7} = \frac{1}{7} \]

### Final Answer:
Thus, the value of \(\frac{\sin \theta - \cos \theta}{\sin \theta + \cos \theta}\) is \(\frac{1}{7}\).

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