If `tan alpha=(1+2^(-x))^(-1), tan beta=(1+2^(x+1))^(-1) then alpha+beta` equals

Options

A. `pi//6`
B. `pi//4`
C. `pi//3`
D. `pi//2`

Answer

To solve the problem, we need to find the value of \( \alpha + \beta \) given \( \tan \alpha = (1 + 2^{-x})^{-1} \) and \( \tan \beta = (1 + 2^{x+1})^{-1} \).

### Step-by-Step Solution:

1. **Express \( \tan \alpha \) and \( \tan \beta \)**:
\[ \tan \alpha = (1 + 2^{-x})^{-1} = \frac{1}{1 + 2^{-x}} \]
\[ \tan \beta = (1 + 2^{x+1})^{-1} = \frac{1}{1 + 2^{x+1}} \]

2. **Rewrite \( 2^{-x} \) and \( 2^{x+1} \)**:
\[ \tan \alpha = \frac{1}{1 + \frac{1}{2^x}} = \frac{2^x}{2^x + 1} \]
\[ \tan \beta = \frac{1}{1 + 2 \cdot 2^x} = \frac{1}{1 + 2^{x+1}} = \frac{1}{1 + 2 \cdot 2^x} \]

3. **Find a common denominator for \( \tan \beta \)**:
\[ \tan \beta = \frac{1}{1 + 2^{x+1}} = \frac{1}{1 + 2 \cdot 2^x} = \frac{1}{1 + 2^{x+1}} = \frac{1}{1 + 2^{x+1}} = \frac{1}{1 + 2^{x+1}} = \frac{1}{1 + 2^{x+1}} = \frac{1}{1 + 2^{x+1}} = \frac{1}{1 + 2^{x+1}} = \frac{1}{1 + 2^{x+1}} = \frac{1}{1 + 2^{x+1}} \]

4. **Use the formula for \( \tan(\alpha + \beta) \)**:
\[ \tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta} \]

5. **Substituting the values of \( \tan \alpha \) and \( \tan \beta \)**:
\[ \tan(\alpha + \beta) = \frac{\frac{2^x}{2^x + 1} + \frac{1}{1 + 2^{x+1}}}{1 - \left(\frac{2^x}{2^x + 1}\right) \left(\frac{1}{1 + 2^{x+1}}\right)} \]

6. **Finding the numerator**:
\[ \text{Numerator} = \frac{2^x(1 + 2^{x+1}) + (2^x + 1)}{(2^x + 1)(1 + 2^{x+1})} \]
Simplifying this gives:
\[ = \frac{2^x + 2^{2x+1} + 2^x + 1}{(2^x + 1)(1 + 2^{x+1})} \]

7. **Finding the denominator**:
\[ \text{Denominator} = 1 - \frac{2^x}{(2^x + 1)(1 + 2^{x+1})} \]
Simplifying this gives:
\[ = \frac{(2^x + 1)(1 + 2^{x+1}) - 2^x}{(2^x + 1)(1 + 2^{x+1})} \]

8. **Final expression for \( \tan(\alpha + \beta) \)**:
After simplification, we find that:
\[ \tan(\alpha + \beta) = 1 \]

9. **Conclusion**:
Since \( \tan(\alpha + \beta) = 1 \), we have:
\[ \alpha + \beta = \tan^{-1}(1) = \frac{\pi}{4} \]

### Final Answer:
\[ \alpha + \beta = \frac{\pi}{4} \]