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: Analyze the given constraint.
\n\nWe are given: \[\n\cos^{-1} \alpha + \cos^{-1} \beta + \cos^{-1} \gamma = 3\pi\n\]\n\nSince \( \cos^{-1} x \) has a range of \( 0 \) to \( \pi \), each term lies between \( 0 \) and \( \pi \).\n\nTherefore, \( \cos^{-1} \alpha, \cos^{-1} \beta, \cos^{-1} \gamma \) must each equal \( \pi \).\n\nThus:\n\[\n\cos^{-1} \alpha = \cos^{-1} \beta = \cos^{-1} \gamma = \pi\n\]\n\nThis leads to:\n\[\n\cos(\pi) = -1\n\quad \Rightarrow \quad\n\alpha = \beta = \gamma = -1\n\]\n\n
Step 2: Evaluate the target expression.
\n\nWe need to find:\n\[\n\alpha(\beta+\gamma) + \beta(\gamma+\alpha) + \gamma(\alpha+\beta)\n\]\n\nSubstitute \( \alpha = \beta = \gamma = -1 \):\n\nFirst term:\n\[\n\alpha(\beta+\gamma) = (-1)((-1)+(-1)) = (-1)(-2) = 2\n\]\n\nSecond term:\n\[\n\beta(\gamma+\alpha) = (-1)((-1)+(-1)) = (-1)(-2) = 2\n\]\n\nThird term:\n\[\n\gamma(\alpha+\beta) = (-1)((-1)+(-1)) = (-1)(-2) = 2\n\]\n\nSumming the terms:\n\[\n2 + 2 + 2 = 6\n\]\n\nSo the answer is \( 6 \).\n\n
Final Answer: \( 6 \).