Question

If a line makes angles α, β and γ with the axes respectively, then cos 2 α + cos 2 β + cos 2 γ =
(a) −2
(b) −1
(c) 1
(d) 2

Answer 1

(b) −1

If a line makes angles α, β and γ with the axes, then
${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1$ ...(1)

We have
$$\begin{gathered} \cos 2\alpha +\cos 2\beta +\cos 2\gamma =2{\cos }^2\alpha -1+2{\cos }^2\beta -1+2{\cos }^2\gamma -1\left[\because \cos 2\theta =2{\cos }^2\theta -1\right] \\ =2\left({\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma \right)-3\left[\mathrm{From}\left(1\right)\right] \\ =2\left(1\right)-3 \\ =-1 \end{gathered}$$