Question

If \( \alpha>\beta>\gamma>0 \), then the expression \[ \cot^{-1} \beta + \left( \frac{1 + \beta^2}{\alpha - \beta} \right) + \cot^{-1} \gamma + \left( \frac{1 + \gamma^2}{\beta - \gamma} \right) + \cot^{-1} \alpha + \left( \frac{1 + \alpha^2}{\gamma - \alpha} \right) \] is equal to:

• A. \( 3\pi \)
• B. \( \frac{\pi}{2} - (\alpha + \beta + \gamma) \)
• C. \( 0 \)
• D. \( \pi \)

Hint:

When dealing with inverse trigonometric functions, use known identities and symmetry properties to simplify the expression. Trigonometric manipulations often help in evaluating such complex expressions.

Answer 1

The Correct Option is D

The given expression involves inverse cotangents and some algebraic manipulation. By applying trigonometric identities and simplifying, we can show that the expression simplifies to \( \pi \). 
Final Answer: \( \pi \).