Question

If a line makes angle $\alpha ,\beta ,\gamma$ with the co-ordinate axes.
Prove that $\cos 2\alpha +\cos 2\beta +\cos 2\gamma +1=0$.

Answer 1

Consider $\cos 2\alpha +\cos 2\beta +\cos 2\gamma +1$

$\cos 2\alpha +\cos 2\beta +\cos 2\gamma +1=2{\cos }^2\alpha -1+2{\cos }^2\beta -1+2{\cos }^2\gamma -1+1$

$=2({\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma )-2$

$=2\left(1\right)-2$

$=2-2$

$=0$

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

Hence proved.