If a line in 3 dimensions makes angles \alpha, \beta and \gamma with the positive directions of the coordinate axes, then what is \cos(\alpha+\beta)\cos(\alpha-\beta) equal to?
- A. \cos^{2}\gamma
- B. -\cos^{2}\gamma ✓
- C. \sin^{2}\gamma
- D. -\sin^{2}\gamma
Correct Answer: B. -\cos^{2}\gamma
Explanation
We know the direction cosines satisfy \cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1. The expression \cos(\alpha+\beta)\cos(\alpha-\beta) simplifies to \cos^2\alpha - \sin^2\beta = \cos^2\alpha - (1 - \cos^2\beta) = \cos^2\alpha + \cos^2\beta - 1. Substituting \cos^2\alpha + \cos^2\beta = 1 - \cos^2\gamma, we get (1 - \cos^2\gamma) - 1 = -\cos^2\gamma.
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