If a line makes angles $\alpha, \beta, \gamma$ 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

When a line in 3D space makes angles with the coordinate axes, these are called direction angles: α with the x-axis, β with the y-axis, and γ with the z-axis.

The fundamental property of direction angles is:

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

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For any angle θ, the basic trigonometric identity states:

$\sin^2\theta + \cos^2\theta = 1$

Therefore:

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

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Applying this identity to each direction angle:

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

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

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

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Adding all three terms:

$\sin^2\alpha + \sin^2\beta + \sin^2\gamma$

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

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

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

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

$\sin^2\alpha + \sin^2\beta + \sin^2\gamma = 3 - 1$

$= 2$

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