If α,β and γ are angle of inclinations of a line with x, y and z axes…

The correct answer is Option (4) → -2

Given: $\alpha, \beta, \gamma$ are the angles of inclination of a line with the x-, y-, and z-axes.

Direction cosines of the line are:

$\cos \alpha$, $\cos \beta$, $\cos \gamma$

Using identity for direction cosines:

$\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$

Use identity: $\cos(2\theta) = 2\cos^2 \theta - 1$

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

$= 2(1) - 3 = -1$

$\Rightarrow 2(\cos(2\alpha) + \cos(2\beta) + \cos(2\gamma)) = 2(-1) = {-2}$