If a line makes an angle $\displaystyle \alpha, \beta, \gamma$ with the coordinate axes, then find the value of $\displaystyle \cos 2 \alpha+\cos 2 \beta+\cos 2 \gamma$.

\[\begin{aligned} & l=\cos \alpha, m=\cos \beta, n=\cos \gamma \Rightarrow \cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1 \\ & \cos 2 \alpha+\cos 2 \beta+\cos 2 \gamma \\ & \quad=2 \cos ^{2} \alpha-1+2 \cos ^{2} \beta-1+2 \cos ^{2} \gamma-1 \\ & \quad=2\left(\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma\right)-3 \\ & \quad=2-3 \\ & \quad=-1 \end{aligned} \]