Question

If a line makes angles $\alpha ,\beta ,\gamma$ with coordinate axes, prove that
$\cos 2\alpha +\cos 2\beta +\cos 2\gamma +1=0$

Answer 1

We Know,

$l=\cos \alpha ,m=\cos \beta ,n=\cos \gamma$

Also,

$l^2+m^2+n^2=1$ ----- $\left(1\right)$

$\therefore co{s}^2\alpha +co{s}^2\beta +co{s}^2\gamma =1$

Considering LHS:

$\cos 2\alpha +\cos 2\beta +\cos 2\gamma +1=0$

$\begin{array}{c}=2{\cos }^2\alpha -1+2{\cos }^2\beta -1+2{\cos }^2\gamma -1+1=0\\ \\ =2\left({\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma \right)-2\\ \\ =2\left(1\right)-2From\left(1\right)\\ \\ =0\end{array}$

Hence Proved.