Question

A line makes angles $\alpha ,\beta ,\gamma$ with the coordinate axes. If $\alpha +\beta =\frac{\pi }{2},$ then $(\cos \alpha +\cos \beta +\cos \gamma {)}^2$ is equal to

• A. $1+\sin 2\alpha$
• B. $1+\cos 2\alpha$
• C. $1-\sin 2\alpha$
• D. $1$

Answer 1

The correct option is A $1+\sin 2\alpha$

Given:$\beta =\frac{\pi }{2}-\alpha$
$\Rightarrow {\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1$
$\Rightarrow {\cos }^2\alpha +{\cos }^2(\frac{\pi }{2}-\alpha )+{\cos }^2\gamma =1$
$\Rightarrow {\cos }^2\alpha +{\sin }^2\alpha +{\cos }^2\gamma =1$
$\Rightarrow 1+{\cos }^2\gamma =1$
$\Rightarrow {\cos }^2\gamma =0$
$\Rightarrow \gamma ={90}^{\circ }$
$(\cos \alpha +\cos \beta +\cos \gamma {)}^2=(\cos \alpha +\sin \alpha +0{)}^2=1+2\sin \alpha \cos \alpha$
$(\cos \alpha +\cos \beta +\cos \gamma {)}^2=1+\sin 2\alpha$