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(D) For a line making angles $\alpha, \beta, \gamma$ with the coordinate axes, the direction cosines are $\cos \alpha, \cos \beta, \cos \gamma$. We kn

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(D) For a line making angles $\alpha, \beta, \gamma$ with the coordinate axes, the direction cosines are $\cos \alpha, \cos \beta, \cos \gamma$. We know that $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$. Step $1$: Calculate $\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = (1 - \cos^2 \alpha) + (1 - \cos^2 \beta) + (1 - \cos^2 \gamma) = 3 - (\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma) = 3 - 1 = 2$. Step $2$: Calculate $\cos 2\alpha + \cos 2\beta + \cos 2\gamma = (2\cos^2 \alpha - 1) + (2\cos^2 \beta - 1) + (2\cos^2 \gamma - 1) = 2(\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma) - 3 = 2(1) - 3 = -1$. Step $3$: The sum is $2 + (-1) = 1$.

If a line makes angles $\alpha, \beta, \gamma$ with the coordinate axes, then the sum of values of $\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma$ and $\cos 2\alpha + \cos 2\beta + \cos 2\gamma$ is ...

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