Question

A lines makes angles $\frac{\alpha }{2},\frac{\beta }{2},\frac{\gamma }{2}$ with positive direction of coordinate axes, then $\cos \alpha +\cos \beta +\cos \gamma$ is equal to

• A. $-1$
• B. $1$
• C. $2$
• D. $3$

Answer 1

The correct option is A $-1$

angles made with $x,y,z$ axis are $\frac{\alpha }{2},\frac{\beta }{2},\frac{\gamma }{2}$

$\therefore \cos {}^2\frac{\alpha }{2}+\cos {}^2\frac{\beta }{2}+\cos {}^2\frac{\gamma }{2}=1\cos 2\theta =2\cos {}^2\theta -1\Rightarrow 1+\cos 2\theta \Rightarrow 2\cos {}^2\frac{\alpha }{2}+2\cos {}^2\frac{\beta }{2}+2\cos {}^2\frac{\gamma }{2}=21+\cos \frac{2\alpha }{2}+1+\cos \frac{2\beta }{2}+1+\cos \frac{2\gamma }{2}=2\cos \alpha +\cos \beta +\cos \gamma =-1$