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?

Question

Question

This question was previously asked in

NDA-I (Mathematics) Official Paper (Held On: 13 Apr, 2025)

Answer 1

Calculation:

We know that $ \cos^{2} \alpha + \cos^{2} \beta + \cos^{2} \gamma = 1 $ ... (i)

And $ \cos(\alpha + \beta) \cdot \cos(\alpha - \beta) = \cos^{2} \alpha - \sin^{2} \beta $

Substitute and simplify:

$ = \cos^{2} \alpha + \cos^{2} \beta - 1 = 1 - \cos^{2} \gamma \quad \Rightarrow \quad - \cos^{2} \gamma $

Hence, the correct answer is Option 2.

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