Question

If a line makes ${\theta }_1,{\theta }_2,{\theta }_3$ angles with the co-ordinates axes, then prove that $\cos 2{\theta }_1+\cos 2{\theta }_2+\cos 2{\theta }_3+1=0$

Answer 1

We know,

${\cos }^2{\theta }_1+{\cos }^2{\theta }_2+{\cos }^2{\theta }_3=1$

$\Rightarrow {2\cos }^2{\theta }_1+{2\cos }^2{\theta }_2+{2\cos }^2{\theta }_3=2$

$\Rightarrow {2\cos }^2{\theta }_1-1+2{\cos }^2{\theta }_2-1+2{\cos }^2{\theta }_3-1=2-1-1-1$

$\Rightarrow {\cos 2\theta }_1+{\cos 2\theta }_2+{\cos 2\theta }_3=-1$

$\Rightarrow {\cos 2\theta }_1+{\cos 2\theta }_2+{\cos 2\theta }_3+1=0$

Proved.