Question

If \( \alpha, \beta, \gamma \) are in AP and \( \tan^{-1}\alpha, \tan^{-1}\beta, \tan^{-1}\gamma \) are also in AP, then

• A. \( \alpha - \beta - \gamma = 0 \)
• B. \( \alpha = \beta = \gamma \)
• C. \( \alpha + \beta = \gamma \)
• D. \( 2\alpha = 3\beta = \gamma \)

Hint:

If both function and inverse function form AP $\longrightarrow$ equality is the safest conclusion.

Answer 1

The Correct Option is B

Concept: If both numbers and their inverse tangent are in AP $\longrightarrow$ only possible when all are equal.

Step 1: Use AP condition.
\[ 2\beta = \alpha + \gamma \]

Step 2: Apply for tan\(^{-1}\).
\[ 2\tan^{-1}\beta = \tan^{-1}\alpha + \tan^{-1}\gamma \]

Step 3: Conclusion.
Only possible when: \[ \alpha = \beta = \gamma \]