If a line makes angles $\alpha$, $\beta$ and $\gamma$ with the positive directions of x-axis, y-axis and z-axis respectively, then $\sin^2\alpha + \sin^2\beta + \sin^2\gamma$ is equal to

A line in 3D space makes angles $\alpha$ with the positive x-axis, $\beta$ with the positive y-axis, and $\gamma$ with the positive z-axis.

When a line makes angles $\alpha$, $\beta$, $\gamma$ with the coordinate axes, the values $\cos \alpha$, $\cos \beta$, $\cos \gamma$ are called the direction cosines of the line.

The direction cosines satisfy the fundamental property:

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

This holds because the direction cosines represent components of a unit vector.

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Using the trigonometric identity $\sin^2\theta + \cos^2\theta = 1$, we can write:

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

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

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

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Adding all three equations:

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

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

$= 3 - 1$

$= 2$

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Therefore, $\sin^2\alpha + \sin^2\beta + \sin^2\gamma = 2$.