Question

If a ray makes angles $\alpha ,\beta ,\gamma$ and $\delta$ with the four diagonals of a cube and
$A:{\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma +{\cos }^2\delta$
$B:{\sin }^2\alpha +{\sin }^2\beta +{\sin }^2\gamma +{\sin }^2\delta$
$C:\cos 2\alpha +\cos 2\beta +\cos 2\gamma +\cos 2\delta$
Arrange $A,B,C$ in descending order

• A. $B,A,C$
• B. $A,B,C$
• C. $C,A,B$
• D. $B,C,A$

Answer 1

The correct option is A $B,A,C$

If a ray makes angles $\alpha ,\beta ,\gamma$ and $\delta$ with the four diagonals of a cube.

Then we know that ${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma +{\cos }^2\delta =\frac{4}{3}$

$A:$

${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma +{\cos }^2\delta =\frac{4}{3}$

$B:$

${\sin }^2\alpha +{\sin }^2\beta +{\sin }^2\gamma +{\sin }^2\delta$

$=1-{\cos }^2\alpha +1-{\cos }^2\beta +1-{\cos }^2\gamma +1-{\cos }^2\delta =4-\frac{4}{3}=\frac{8}{3}$

$C:$

$\cos 2\alpha +\cos 2\beta +\cos 2\gamma +\cos 2\delta$

$=1-2{\sin }^2\alpha +1-2{\sin }^2\beta +1-2{\sin }^2\gamma +1-2{\sin }^2\delta =4-\frac{16}{3}=-\frac{4}{3}$

Descending order is $B,A,C$

Hence, option A is correct answer.