If `b tan theta ` = a find the value of `(cos theta + sin theta )/( cos theta - sin theta)`

Answer

To solve the problem, we need to find the value of \(\frac{\cos \theta + \sin \theta}{\cos \theta - \sin \theta}\) given that \(b \tan \theta = a\).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ b \tan \theta = a \]
From this, we can express \(\tan \theta\) in terms of \(a\) and \(b\):
\[ \tan \theta = \frac{a}{b} \]

2. **Rewrite the expression we need to evaluate:**
We need to find:
\[ \frac{\cos \theta + \sin \theta}{\cos \theta - \sin \theta} \]

3. **Divide the numerator and denominator by \(\cos \theta\):**
\[ \frac{\cos \theta + \sin \theta}{\cos \theta - \sin \theta} = \frac{\frac{\cos \theta}{\cos \theta} + \frac{\sin \theta}{\cos \theta}}{\frac{\cos \theta}{\cos \theta} - \frac{\sin \theta}{\cos \theta}} = \frac{1 + \tan \theta}{1 - \tan \theta} \]

4. **Substitute \(\tan \theta\) with \(\frac{a}{b}\):**
Now, replace \(\tan \theta\) in the expression:
\[ \frac{1 + \tan \theta}{1 - \tan \theta} = \frac{1 + \frac{a}{b}}{1 - \frac{a}{b}} \]

5. **Simplify the expression:**
To simplify, find a common denominator for both the numerator and the denominator:
\[ = \frac{\frac{b + a}{b}}{\frac{b - a}{b}} = \frac{b + a}{b - a} \]

6. **Final Result:**
Thus, the value of \(\frac{\cos \theta + \sin \theta}{\cos \theta - \sin \theta}\) is:
\[ \frac{b + a}{b - a} \]