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Question
If a line makes angles α, β and γ with the coordinate axes, find the value of cos2α + cos2β + cos2γ.
Solution
\[ \text{ It is given that the the line makes angles } \alpha, \beta, \gamma \text{ with the coordinate axis }. \]
\[ \therefore l = \cos \alpha, m = \cos \beta \text{ and } n = \cos \gamma\]
\[ \Rightarrow l^2 + m^2 + n^2 = 1\]
\[ \Rightarrow \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 . . . \left( 1 \right)\]
\[\text{ Now} , \]
\[\cos^2\alpha + \cos^2\beta + \cos^2\gamma = \left( 2 \cos^2 \alpha - 1 \right) + \left( 2 \cos^2 \beta - 1 \right) + \left( 2 \cos^2 \gamma - 1 \right)\]
\[ = 2\left( \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma \right) - 3\]
\[ = 2\left( 1 \right) - 3 ............\left [ \text{ From }\left( 1\right) \right]\]
\[ = - 1\]
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