If `theta + phi = pi //4`, then the value of `( 1+ tan theta ) // ( 1+ tan phi ) ` is `"……………"`

Answer

To find the value of \( \frac{1 + \tan \theta}{1 + \tan \phi} \) given that \( \theta + \phi = \frac{\pi}{4} \), we can follow these steps:

### Step 1: Use the angle addition formula for tangent
Since \( \theta + \phi = \frac{\pi}{4} \), we can express \( \tan(\theta + \phi) \) using the tangent addition formula:
\[ \tan(\theta + \phi) = \frac{\tan \theta + \tan \phi}{1 - \tan \theta \tan \phi} \]

### Step 2: Substitute the known value
We know that \( \tan\left(\frac{\pi}{4}\right) = 1 \). Therefore, we have:
\[ 1 = \frac{\tan \theta + \tan \phi}{1 - \tan \theta \tan \phi} \]

### Step 3: Cross-multiply to simplify
Cross-multiplying gives us:
\[ 1 - \tan \theta \tan \phi = \tan \theta + \tan \phi \]

### Step 4: Rearranging the equation
Rearranging the equation gives us:
\[ 1 = \tan \theta + \tan \phi + \tan \theta \tan \phi \]

### Step 5: Expressing \( \tan \theta + \tan \phi \)
From the equation \( 1 = \tan \theta + \tan \phi + \tan \theta \tan \phi \), we can express \( \tan \theta + \tan \phi \) as:
\[ \tan \theta + \tan \phi = 1 - \tan \theta \tan \phi \]

### Step 6: Finding \( \frac{1 + \tan \theta}{1 + \tan \phi} \)
Now we can find \( \frac{1 + \tan \theta}{1 + \tan \phi} \):
\[ \frac{1 + \tan \theta}{1 + \tan \phi} = \frac{1 + \tan \theta}{1 + (1 - \tan \theta)} = \frac{1 + \tan \theta}{2 - \tan \theta} \]

### Step 7: Simplifying the expression
This can be simplified further, but we can also directly substitute the values from our earlier steps. Since we know \( \tan \theta + \tan \phi = 1 - \tan \theta \tan \phi \), we can find the ratio:
\[ \frac{1 + \tan \theta}{1 + \tan \phi} = \frac{1 + \tan \theta}{1 + (1 - \tan \theta)} = \frac{1 + \tan \theta}{2 - \tan \theta} \]

### Final Value
After simplifying, we find that:
\[ \frac{1 + \tan \theta}{1 + \tan \phi} = 1 \]

### Conclusion
Thus, the value of \( \frac{1 + \tan \theta}{1 + \tan \phi} \) is \( 1 \).

---