If a line makes angles $\alpha, \beta, \gamma$ with the positive directions of the coordinate axes, then the value of $\cos 2\alpha + \cos 2\beta + \cos 2\gamma$ is

When a line makes angles $\alpha, \beta, \gamma$ with the positive directions of the x-axis, y-axis, and z-axis respectively, the values $\cos \alpha, \cos \beta, \cos \gamma$ are the direction cosines of that line.

For any line in 3D space, the direction cosines satisfy:

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

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To find the value of $\cos 2\alpha + \cos 2\beta + \cos 2\gamma$, apply the double angle formula:

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

Applying this to each angle:

$\cos 2\alpha = 2\cos^2 \alpha - 1$

$\cos 2\beta = 2\cos^2 \beta - 1$

$\cos 2\gamma = 2\cos^2 \gamma - 1$

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Adding the three expressions:

$\cos 2\alpha + \cos 2\beta + \cos 2\gamma$

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

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

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

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Using the fundamental property $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$:

$= 2(1) - 3$

$= -1$

Therefore, $\cos 2\alpha + \cos 2\beta + \cos 2\gamma = -1$