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 addition formula to simplify expressions with angle sums. This approach helps in precisely finding values for individual angles in trigonometric identities.

Answer 1

The Correct Option is C

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 \( \boxed{\frac{1}{\sqrt{3}}} \).