If a line makes angles \(\alpha, \beta, \gamma\) with the positive directions of the coordinate axes, then the value of \(\cos 2\alpha + \cos 2\beta + \cos 2\gamma\) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → -1
Here \(\cos\alpha, \cos\beta, \cos\gamma\) are the direction cosines of the line, so
\(\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1\)
Using \(\cos 2\theta = 2\cos^2\theta - 1\) for each angle:
\(\cos 2\alpha + \cos 2\beta + \cos 2\gamma = 2(\cos^2\alpha + \cos^2\beta + \cos^2\gamma) - 3\)
\(= 2(1) - 3 = -1\)
Shortcut: Take the line to be the x-axis, so \(\alpha = 0\) and \(\beta = \gamma = \frac{\pi}{2}\):
\(\cos 0 + \cos\pi + \cos\pi = 1 - 1 - 1 = -1\)