Question

If $\cos \alpha ,\cos \beta ,\cos \gamma$ are direction-cosines of any straight line, then prove that:
$\cos 2\alpha +\cos 2\beta +\cos 2\gamma =-1$

Answer 1

$\cos \alpha ,\cos \beta ,\cos \gamma$ are direction-cosines of a straight line
$\therefore {\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1$
now from
$L.H.S.=\cos 2\alpha +\cos 2\beta +\cos 2\gamma$
$=2{\cos }^2\alpha -1+2{\cos }^2\beta -1+2{\cos }^2\gamma -1$
$=2({\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma )-3$
$=2\times 1-3$
$=2-3$
$=-1=R.H.S.$
$L.H.S=R.H.S$