Question

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

• A. $-1$
• B. $0$
• C. $1$
• D. $2$

Answer 1

The correct option is C

$1$

Explanation for the correct option:

Find the value of given expression:

Given, The line makes angles $\alpha ,\beta$ and $\gamma$ with coordinate axis

$\therefore l^2+m^2+n^2=1$

$\Rightarrow$${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1$ ….(1)

Now,

$\begin{array}{rcl}\cos 2\alpha +\cos 2\beta +\cos 2\gamma +{\sin }^2\alpha +{\sin }^2\beta +{\sin }^2\gamma & =& \left(2{\cos }^2\alpha -1\right)+\left(2{\cos }^2\beta -1\right)+\left(2{\cos }^2\gamma -1\right)+{\sin }^2\alpha +{\sin }^2\beta +{\sin }^2\gamma \\ & =& {\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma \\ & =& \mathbf{1}\end{array}$

Hence, Option ‘C’ is Correct.