Mathematics - 3D Coordinate Geometry Question with Solution | TestHub

The sum to be evaluated is ${\sin }^2\alpha +{\sin }^2\beta +{\sin }^2\gamma +\cos 2\alpha +\cos 2\beta +\cos 2\gamma$.

Using the trigonometric identity $\cos 2\theta ={\cos }^2\theta -{\sin }^2\theta$, we can write ${\sin }^2\theta +\cos 2\theta ={\cos }^2\theta$.

Grouping the terms, we get:

$({\sin }^2\alpha +\cos 2\alpha )+({\sin }^2\beta +\cos 2\beta )+({\sin }^2\gamma +\cos 2\gamma )$

$={\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma$

Since $\alpha ,\beta ,\gamma$ are the angles made by a line with the coordinate axes, their direction cosines satisfy:

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

Thus, the required sum is $1$.

Answer: $1$