Question

If a line makes angles $\alpha ,\beta ,\gamma$ with axes of co-ordinates, then $\cos 2\alpha +\cos 2\beta +\cos 2\gamma$ is equla to

• A. $-2$
• B. $-1$
• C. $1$
• D. $2$

Answer 1

The correct option is B $-1$

We know that direction cosines are $\cos \alpha ,\cos \beta ,\cos \gamma$

Then sum of the squares of the direction cosine is 1

$i.e.{\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1$

we know Trignometric identity that ${\sin }^{2}A+{\cos }^{2}A=1$

From the above identity we can write as ${\cos }^{2}A=1-{\sin }^{2}A$

Now replacing we get $(1-{\sin }^2\alpha )+(1-{\sin }^2\beta )+(1-{\sin }^2\gamma )$

By adding all we get $3-({\sin }^2\alpha +{\sin }^2\beta +{\sin }^2\gamma )=1$

$\therefore \cos 2\alpha +\cos 2\beta +\cos 2\gamma =2-3=-1$