Question

If \( \tan(\alpha + \beta) = \sqrt{3} \) and \( \tan \alpha = \frac{1}{\sqrt{3}} \), then the value of \( \tan \beta \) is:

• A. \( \frac{1}{6} \)
• B. \( \frac{1}{7} \)
• C. \( \frac{1}{\sqrt{3}} \)
• D. \( \frac{7}{6} \)

Hint:

Use the tan sum formula to solve such problems.

Answer 1

The Correct Option is D

Using tan sum formula:
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \] Substituting values:
\[ \sqrt{3} = \frac{\frac{1}{\sqrt{3}} + \tan \beta}{1 - \frac{1}{\sqrt{3}} \cdot \tan \beta} \] Solving for \( \tan \beta \):
\[ \tan \beta = \frac{7}{6} \]