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Question
If a line makes angles α, β and γ with the axes respectively, then cos 2 α + cos 2 β + cos 2 γ =
Options
−2
−1
1
2
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Solution
−1
If a line makes angles α, β and γ with the axes, then
\[\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1\]
We have ,
\[\cos 2\alpha + \cos 2\beta + \cos 2\gamma = 2 \cos^2 \alpha - 1 + 2 \cos^2 \beta - 1 + 2 \cos^2 \gamma - 1 \left[ \because \cos 2\theta = 2 \cos^2 \theta - 1 \right]\]
\[ = 2 \left( \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma \right) - 3 \left[ \text { From } \left( 1 \right) \right]\]
\[ = 2\left( 1 \right) - 3\]
\[ = - 1\]
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