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: \( \pi \).

Answer 2

Step 1: Understand the given expression.
Consider the expression:
\[ \cot^{-1} \beta + \frac{1 + \beta^2}{\alpha - \beta} + \cot^{-1} \gamma + \frac{1 + \gamma^2}{\beta - \gamma} + \cot^{-1} \alpha + \frac{1 + \alpha^2}{\gamma - \alpha} \] with the condition \( \alpha > \beta > \gamma > 0 \).

Step 2: Identify patterns involving inverse cotangent sums.
The terms involving \(\cot^{-1}\) and the fractions correspond to sums and differences of inverse cotangent functions combined with rational expressions.

Step 3: Apply known inverse cotangent identities.
Using the identity:
\[ \cot^{-1} x + \cot^{-1} y = \cot^{-1} \left( \frac{xy - 1}{x + y} \right) \] and algebraic manipulation, the expression simplifies.

Step 4: Final simplification.
Considering the order \( \alpha > \beta > \gamma > 0 \), the expression evaluates neatly to:
\[ \boxed{\pi} \] This is a known result in trigonometric identities involving inverse cotangent sums and rational functions.