If a line makes angles α, β, γ with the axis then cos 2α + cos 2β + cos 2γ =

The correct option is -1.

Let us understand why this is the correct answer step-by-step.
We are given that a line makes angles $\alpha$, $\beta$, and $\gamma$ with the coordinate axes.
The direction cosines of this line are given by:
$l=\cos \alpha$, $m=\cos \beta$, and $n=\cos \gamma$.

A fundamental property of direction cosines for any line in three-dimensional space is that the sum of their squares is always equal to 1:

$l^2+m^2+n^2=1$

Substituting the values of $l$, $m$, and $n$, we get:

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

We need to find the value of:
$S=\cos 2\alpha +\cos 2\beta +\cos 2\gamma$

Recall the trigonometric double-angle identity for cosine:

$\cos 2\theta =2{\cos }^2\theta -1$

Applying this identity to each term in our expression:
$\cos 2\alpha =2{\cos }^2\alpha -1$
$\cos 2\beta =2{\cos }^2\beta -1$
$\cos 2\gamma =2{\cos }^2\gamma -1$

Now, substitute these identities back into the expression for $S$:

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

Grouping the terms:

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

Substitute the direction cosines identity ${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1$ into the equation:

$S=2\left(1\right)-3$

Simplifying the arithmetic:

$S=2-3=-1$

Thus, the value of $\cos 2\alpha +\cos 2\beta +\cos 2\gamma$ is equal to -1.