Comment
Share
Q)

# A line makes angle $\theta$ with both $x\:axis$ and $z\:axis$. If it makes angle $\alpha$ with $y\:axis$ and $\sin^2\alpha=2 \sin^2\theta$ then $\cot^2\theta=?$

$(a)\;2/3\:\:\:\qquad\:\:(b)\:\:1\:\:\:\qquad\:\:(c)\:\:3/5\:\:\:\qquad\:\:(d)\:\:5/3$

Since the line makes angle $\theta,\alpha\:and\:\theta$ with the coordinate axes,
$cos^2\theta+cos^2\alpha+cos^2\theta=1$
$\Rightarrow\:2cos^2\theta+cos^2\alpha=1$....(i)
Given: $sin^2\alpha=2sin^2\theta$
$\Rightarrow\:1-cos^2\alpha=2(1-cos^2\theta)$
$\Rightarrow\:cos^2 \alpha=2cos^2\theta-1$.....(ii)
Solving (i) and (ii), $cos^2\theta=\large\frac{1}{2}$
$\Rightarrow\:cot^2\theta=1$