Question

If a line in the space makes angles $\alpha ,\beta$ and $\gamma$ with the coordinate axes, then $\cos 2\alpha +\cos 2\beta +\cos 2\gamma +{\sin }^2\alpha +{\sin }^2\beta +{\sin }^2\gamma =$

• A. $-1$
• B. $0$
• C. $1$
• D. $2$

Answer 1

The correct option is A $-1$

As $\cos 2\alpha =1-2{\sin }^2\alpha ,$ similarly for $\cos 2\beta$ and $\cos 2\gamma$ expression becomes $1-({\sin }^2\alpha +{\sin }^2\beta +{\sin }^2\gamma )$. Now ${\sin }^2\alpha =1-{\cos }^2\alpha$

Similarly for ${\sin }^2\beta ,{\sin }^2\gamma$, expression becomes, $1-(3-\sum {\cos }^2\alpha )=\sum {\cos }^2\alpha -2$, as $\sum {\cos }^2\alpha =1$

$\Rightarrow$ Expression $=-1$

$\Rightarrow \left(A\right)$.