Question

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

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

Hint:

When dealing with inverse trigonometric equations involving sums of inverse cosines, check if each term attains boundary values like \( 0 \) or \( \pi \) for easy simplification.

Answer 1

The Correct Option is C

Step 1: Understand the given condition.
We are given: \[ \cos^{-1} \alpha + \cos^{-1} \beta + \cos^{-1} \gamma = 3\pi \] Since the range of \( \cos^{-1} x \) is from \( 0 \) to \( \pi \), each \( \cos^{-1} \) term individually lies between \( 0 \) and \( \pi \). Thus, \( \cos^{-1} \alpha, \cos^{-1} \beta, \cos^{-1} \gamma \) must each be \( \pi \). Hence: \[ \cos^{-1} \alpha = \cos^{-1} \beta = \cos^{-1} \gamma = \pi \] From this: \[ \cos(\pi) = -1 \quad \Rightarrow \quad \alpha = \beta = \gamma = -1 \]
Step 2: Find the required expression.
We are asked to find: \[ \alpha(\beta+\gamma) + \beta(\gamma+\alpha) + \gamma(\alpha+\beta) \] Substituting \( \alpha = \beta = \gamma = -1 \): First term: \[ \alpha(\beta+\gamma) = (-1)((-1)+(-1)) = (-1)(-2) = 2 \] Second term: \[ \beta(\gamma+\alpha) = (-1)((-1)+(-1)) = (-1)(-2) = 2 \] Third term: \[ \gamma(\alpha+\beta) = (-1)((-1)+(-1)) = (-1)(-2) = 2 \] Adding: \[ 2 + 2 + 2 = 6 \] Thus, the value is \( 6 \).
Final Answer: \( 6 \).