Question

If \( \cos^{-1}\alpha + \cos^{-1}\beta + \cos^{-1}\gamma = 3\pi \), then \( \alpha(\beta+\gamma) + \beta(\gamma+\alpha) + \gamma(\alpha+\beta) \) is equal to:

• A. \( 0 \)
• B. \( 1 \)
• C. \( 6 \)
• D. \( 12 \)

Hint:

For inverse trigonometric sums: Always recall principal value ranges. If sum equals maximum possible value, each term must be maximal. \( \cos^{-1}x = \pi \Rightarrow x = -1 \)

Answer 1

The Correct Option is C

Concept:
The principal value range of the inverse cosine function is: \[ 0 \le \cos^{-1}x \le \pi \] So the maximum possible value of each term is \( \pi \).

If a sum of multiple inverse cosine terms equals the maximum possible total, then each term must individually be at its maximum.

Step 1: Use the range of inverse cosine.
Given: \[ \cos^{-1}\alpha + \cos^{-1}\beta + \cos^{-1}\gamma = 3\pi \] Since each term \( \le \pi \), equality is possible only if: \[ \cos^{-1}\alpha = \cos^{-1}\beta = \cos^{-1}\gamma = \pi \]

Step 2: Find values of \( \alpha, \beta, \gamma \).
\[ \cos^{-1}x = \pi \Rightarrow x = \cos \pi = -1 \] Hence: \[ \alpha = \beta = \gamma = -1 \]

Step 3: Evaluate the expression.
\[ \alpha(\beta+\gamma) + \beta(\gamma+\alpha) + \gamma(\alpha+\beta) \] Substitute \( \alpha = \beta = \gamma = -1 \): \[ (-1)((-1)+(-1)) + (-1)((-1)+(-1)) + (-1)((-1)+(-1)) \] Each term: \[ (-1)(-2) = 2 \] So total: \[ 2 + 2 + 2 = 6 \]