If a line makes an angle $\alpha, \beta, \gamma$ with the coordinate axis, then find the value of $\cos 2\alpha + \cos 2\beta + \cos 2\gamma$.

$-1$

The correct answer is Option (3) → $-1$ ##

We have, $\cos 2\alpha + \cos 2\beta + \cos 2\gamma $

$= (2\cos^2 \alpha - 1) + (2\cos^2 \beta - 1) + (2\cos^2 \gamma - 1)$

$= 2(\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma) – 3$

$= 2(1) - 3 \quad [∵\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1]$

$= 2 – 3$

$= -1$