The following is the question in my textbook:-
If a straight line makes angle $\alpha$, $\beta$, $\gamma$ with the $x, y, z$ axes respectively, then show that $\sin^2{\alpha} + \sin^2{\beta} + \sin^2{\gamma} = 2 $?
Here is what I have done:-
Since $\alpha$, $\beta$, $\gamma$ are the angles made by the line with the axes so $\cos\alpha$, $\cos\beta$, $\cos\gamma$ are the direction cosines of the line
now
$$\sin^2\theta + \cos^2\theta = 1$$
$$\sin^2\theta = 1 - \cos^2\theta$$
so $$\sin^2\alpha + \sin^2\beta + \sin^2\gamma = 1 - \cos^2\alpha + 1 - \cos^2\beta + 1 - \cos^2\gamma$$$$ =3 - (\cos^2\alpha + \cos^2\beta + \cos^2\gamma)$$ now since $\cos\alpha$, $\cos\beta$, $\cos\gamma$ are the direction cosines of the line so $\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1$
so $$3 - \cos^2\alpha + \cos^2\beta + \cos^2\gamma= 3 - 1$$ $$=2- answer$$
My question was that have I done it correctly and if not what is the correct way of doing it.