If \(\alpha, \beta\) and \(\gamma\) are angle of inclinations of a line with x, y and z axes respectively, then the value of \(2(\cos 2\alpha + \cos 2\beta + \cos 2\gamma)\) is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → −2
The direction cosines of the line are \(\cos\alpha, \cos\beta, \cos\gamma\), and
\(\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1\)
Using \(\cos 2\theta = 2\cos^2\theta - 1\):
\(\cos 2\alpha + \cos 2\beta + \cos 2\gamma = 2(\cos^2\alpha + \cos^2\beta + \cos^2\gamma) - 3\)
\(= 2(1) - 3 = -1\)
So
\(2(\cos 2\alpha + \cos 2\beta + \cos 2\gamma) = 2(-1) = -2\)