If a line makes angles \(\alpha,\beta,\gamma\) with the positive directions of x-axis, y-axis, z-axis respectively, then the value of \(\cos 2\alpha + \cos 2\beta+ \cos 2\gamma\) is equal to
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → -1
The direction cosines \(\cos\alpha, \cos\beta, \cos\gamma\) of a line satisfy
\(\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1\)
Using the double-angle formula \(\cos 2\theta = 2\cos^2\theta - 1\) for each angle:
\(\cos 2\alpha + \cos 2\beta + \cos 2\gamma = 2(\cos^2\alpha + \cos^2\beta + \cos^2\gamma) - 3\)
\(= 2(1) - 3 = -1\)