NDA 2025 · PYQ · 3D Geometry / Direction Cosines · medium
If a line in 3 dimensions makes angles α, β and γ with the positive directions of the coordinate axes, then what is cos(α+β)cos(α-β) equal to?
Answer
The correct answer is B: -cos²γ. cos(α+β)cos(α-β) = cos²α - sin²β. Using cos²α + cos²β + cos²γ = 1: cos²α = 1 - cos²β - cos²γ.
- A.cos²γ
- B.-cos²γ✓ Correct
- C.sin²γ
- D.-sin²γ
cos(α+β)cos(α-β) = cos²α - sin²β. Using cos²α + cos²β + cos²γ = 1: cos²α = 1 - cos²β - cos²γ. So cos²α - sin²β = 1 - cos²β - cos²γ - (1-cos²β) = -cos²γ.
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