If `(1+tantheta)(1+tanphi)=2`, then `theta + phi`=

Options

A. `30^(@)`
B. `45^(@)`
C. `60^(@)`
D. `75^(@)`

Answer

To solve the equation \((1 + \tan \theta)(1 + \tan \phi) = 2\), we will follow these steps:

### Step 1: Expand the left-hand side
We start by expanding the expression:
\[ (1 + \tan \theta)(1 + \tan \phi) = 1 + \tan \theta + \tan \phi + \tan \theta \tan \phi \]
Setting this equal to 2, we have:
\[ 1 + \tan \theta + \tan \phi + \tan \theta \tan \phi = 2 \]

### Step 2: Rearrange the equation
Now, we can rearrange the equation:
\[ \tan \theta + \tan \phi + \tan \theta \tan \phi = 2 - 1 \]
This simplifies to:
\[ \tan \theta + \tan \phi + \tan \theta \tan \phi = 1 \]

### Step 3: Use the tangent addition formula
Recall the tangent addition formula:
\[ \tan(\theta + \phi) = \frac{\tan \theta + \tan \phi}{1 - \tan \theta \tan \phi} \]
Using this, we can express our equation as:
\[ \tan(\theta + \phi) = \frac{\tan \theta + \tan \phi}{1 - \tan \theta \tan \phi} \]

### Step 4: Substitute values
From our earlier result, we know:
\[ \tan \theta + \tan \phi = 1 - \tan \theta \tan \phi \]
Let \(x = \tan \theta + \tan \phi\) and \(y = \tan \theta \tan \phi\). Then we can rewrite:
\[ \tan(\theta + \phi) = \frac{x}{1 - y} \]
Substituting \(x = 1 - y\):
\[ \tan(\theta + \phi) = \frac{1 - y}{1 - y} \]
This simplifies to:
\[ \tan(\theta + \phi) = 1 \]

### Step 5: Find the angle
The equation \(\tan(\theta + \phi) = 1\) implies:
\[ \theta + \phi = \tan^{-1}(1) \]
The value of \(\tan^{-1}(1)\) is:
\[ \theta + \phi = \frac{\pi}{4} \quad \text{or} \quad 45^\circ \]

### Final Answer
Thus, the final answer is:
\[ \theta + \phi = \frac{\pi}{4} \quad \text{or} \quad 45^\circ \]

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