#### Question

If a line makes angles α, β and γ with the coordinate axes, find the value of cos 2α + cos 2β + cos 2γ.

#### Solution

\[ \text{ It is given that the the line makes angles } \alpha, \beta, \gamma \text{ with the coordinate axis }. \]

\[ \therefore l = \cos \alpha, m = \cos \beta \text{ and } n = \cos \gamma\]

\[ \Rightarrow l^2 + m^2 + n^2 = 1\]

\[ \Rightarrow \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 . . . \left( 1 \right)\]

\[\text{ Now} , \]

\[\cos 2\alpha + \cos 2\beta + \cos 2\gamma = \left( 2 \cos^2 \alpha - 1 \right) + \left( 2 \cos^2 \beta - 1 \right) + \left( 2 \cos^2 \gamma - 1 \right)\]

\[ = 2\left( \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma \right) - 3\]

\[ = 2\left( 1 \right) - 3 \left [ \text{ From }\left( 1\right) \right]\]

\[ = - 1\]