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 addition formula to simplify expressions with angle sums. This approach helps in precisely finding values for individual angles in trigonometric identities.
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 C
Solution and Explanation
Step 1: Use the tan addition formula:
\[
\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \cdot \tan \beta}
\]
Step 2: Substituting the given values:
\[
\sqrt{3} = \frac{\frac{1}{\sqrt{3}} + \tan \beta}{1 - \frac{1}{\sqrt{3}} \cdot \tan \beta}
\]
Step 3: Simplify the equation and solve for \( \tan \beta \):
\[
\tan \beta = \frac{1}{\sqrt{3}}
\]
Thus, the correct answer is \( \boxed{\frac{1}{\sqrt{3}}} \).
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