Question

If $ \tan x = \frac{m}{m+1} $, $ \tan \beta = \frac{1}{2m+1} $, then $ (\alpha + \beta) $ is equal to

• A. \( \frac{\pi}{2} \)
• B. \( \frac{\pi}{4} \)
• C. \( \frac{\pi}{6} \)
• D. None of these

Hint:

When using the addition formula for tangents, simplify both the numerator and denominator carefully.

Answer 1

The Correct Option is B

Step 1: Use the Addition Formula for Tangents
We are given: \[ \tan \alpha = \frac{m}{m+1}, \quad \tan \beta = \frac{1}{2m+1} \] We can use the addition formula for tangents: \[ \tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta} \]
Step 2: Substitute the Values
Substituting the given values: \[ \tan(\alpha + \beta) = \frac{\frac{m}{m+1} + \frac{1}{2m+1}}{1 - \frac{m}{m+1} \cdot \frac{1}{2m+1}} \] Simplify the numerator and denominator: \[ \tan(\alpha + \beta) = 1 \] Thus, \( \alpha + \beta = \frac{\pi}{4} \).
Step 3: Conclusion Thus, \( (\alpha + \beta) = \frac{\pi}{4} \).