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A line makes the same angle $\alpha$ with each of the $x$ and $y$ axes. If the angle $\theta$, which it makes with the $z$ -axis, is such that $\sin ^{2} \theta=2 \sin ^{2} \alpha$, then what is the value of $\alpha$ ?
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The correct answer is:
$\pi / 4$
Since $l^{2}+m^{2}+n^{2}=1$
$\therefore \cos ^{2} \alpha+\cos ^{2} \alpha+\cos ^{2} \theta=1$
$(\because A$ line makes the same angle $\alpha$ with $x$ and $y$ -axes and $\theta$ with z-axis) Also, $\sin ^{2} \theta=2 \sin ^{2} \alpha$
$\Rightarrow 1-\cos ^{2} \theta=2\left(1-\cos ^{2} \alpha\right)\left(\because \sin ^{2} A+\cos ^{2} A=1\right)$
$\Rightarrow \cos ^{2} \theta=2 \cos ^{2} \alpha-1$
From Eq. (i) and (ii) $2 \cos ^{2} \alpha+2 \cos ^{2} \alpha-1=1$
$\Rightarrow 4 \cos ^{2} \alpha=2 \Rightarrow \cos ^{2} \alpha=\frac{1}{2}$
$\Rightarrow \cos \alpha=\pm \frac{1}{\sqrt{2}}$
$\Rightarrow \alpha=\frac{\pi}{4}, \frac{3 \pi}{4}$
$\therefore \cos ^{2} \alpha+\cos ^{2} \alpha+\cos ^{2} \theta=1$
$(\because A$ line makes the same angle $\alpha$ with $x$ and $y$ -axes and $\theta$ with z-axis) Also, $\sin ^{2} \theta=2 \sin ^{2} \alpha$
$\Rightarrow 1-\cos ^{2} \theta=2\left(1-\cos ^{2} \alpha\right)\left(\because \sin ^{2} A+\cos ^{2} A=1\right)$
$\Rightarrow \cos ^{2} \theta=2 \cos ^{2} \alpha-1$
From Eq. (i) and (ii) $2 \cos ^{2} \alpha+2 \cos ^{2} \alpha-1=1$
$\Rightarrow 4 \cos ^{2} \alpha=2 \Rightarrow \cos ^{2} \alpha=\frac{1}{2}$
$\Rightarrow \cos \alpha=\pm \frac{1}{\sqrt{2}}$
$\Rightarrow \alpha=\frac{\pi}{4}, \frac{3 \pi}{4}$
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