NDA 1 2025 Maths, question 61

Topic: Analytical geometry

Q61

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) $- \cos^{2}\gamma$ Correct answer
(c) $\sin^{2}\gamma$
(d) $- \sin^{2}\gamma$

Solution.

Answer: (b) $- \cos^{2}\gamma$

Use $\cos(\alpha + \beta)\cos(\alpha - \beta) = \cos^{2}\alpha - \sin^{2}\beta = \cos^{2}\alpha + \cos^{2}\beta - 1$ .
Direction cosines satisfy $\cos^{2}\alpha + \cos^{2}\beta + \cos^{2}\gamma = 1$ .
Therefore the expression is $- \cos^{2}\gamma$ .