If a line makes angles α,β,γ with the positive directions of the coor…

The correct answer is Option (3) → -1

Let a line make angles $\alpha$, $\beta$, and $\gamma$ with the positive directions of the coordinate axes.

Then the direction cosines of the line are:

$l = \cos \alpha$, $m = \cos \beta$, $n = \cos \gamma$

Using the identity:

$\cos(2\theta) = 2\cos^2\theta - 1$

So,

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

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

Using the direction cosine identity:

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

Therefore,

$\cos(2\alpha) + \cos(2\beta) + \cos(2\gamma) = 2(1) - 3 = -1$