Question:
If \( \tan(\alpha + \beta) = \sqrt{3} \) and \( \tan \alpha = \frac{1}{\sqrt{3}} \), then the value of \( \tan \beta \) is:
If \( \tan(\alpha + \beta) = \sqrt{3} \) and \( \tan \alpha = \frac{1}{\sqrt{3}} \), then the value of \( \tan \beta \) is:
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Use the tan sum formula to solve such problems.
Updated On: Oct 27, 2025
- \( \frac{1}{6} \)
- \( \frac{1}{7} \)
- \( \frac{1}{\sqrt{3}} \)
- \( \frac{7}{6} \)
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The Correct Option is D
Solution and Explanation
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 \( \tan \beta \):
\[ \tan \beta = \frac{7}{6} \]
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \] Substituting values:
\[ \sqrt{3} = \frac{\frac{1}{\sqrt{3}} + \tan \beta}{1 - \frac{1}{\sqrt{3}} \cdot \tan \beta} \] Solving for \( \tan \beta \):
\[ \tan \beta = \frac{7}{6} \]
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