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  4. If cos α, cos β, cos γ, are direction cosine of a line then value of sin2 α + sin2 β + sin2 γ is:

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Q. If cos $\alpha$, cos $\beta$, cos $\gamma$, are direction cosine of a line then value of $\sin^2 \, \alpha + \sin^2 \, \beta + \sin^2 \gamma$ is:

Three Dimensional Geometry

Solution:

If a line OP makes angles $\alpha$, $\beta$, $\gamma$ respectively with x, y, z axes, the direction cosines are $\cos \alpha, \cos \beta, \cos \gamma$.
Then, $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$
Consider $\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 $