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A line makes the same angle ' $\alpha$ ' with each of the $\mathrm{x}$ and $\mathrm{y}$ axes. If the angle ' $\theta$ ', which it makes with the z-axis, is such that $\sin ^2 \theta=2 \sin ^2 \alpha$, then the angle $\alpha$ is
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The correct answer is:
$\left(\frac{\pi}{4}\right)$
Direction cosines of the line are $\cos ^2 \alpha+\cos ^2 \alpha+\cos ^2 \theta=1$
$\begin{aligned} & \Rightarrow \cos ^2 \alpha+\cos ^2 \alpha+1-\sin ^2 \theta=1 \\ & \Rightarrow \cos ^2 \alpha+\cos ^2 \alpha+1-2 \sin ^2 \alpha=1 \\ & \Rightarrow 1-\sin ^2 \alpha+1-\sin ^2 \alpha+1-2 \sin ^2 \alpha=1 \\ & \Rightarrow \sin ^2 \alpha=\frac{1}{2}=\sin ^2 \frac{\pi}{4} \\ & \Rightarrow \alpha=\frac{\pi}{4}\end{aligned}$
$\begin{aligned} & \Rightarrow \cos ^2 \alpha+\cos ^2 \alpha+1-\sin ^2 \theta=1 \\ & \Rightarrow \cos ^2 \alpha+\cos ^2 \alpha+1-2 \sin ^2 \alpha=1 \\ & \Rightarrow 1-\sin ^2 \alpha+1-\sin ^2 \alpha+1-2 \sin ^2 \alpha=1 \\ & \Rightarrow \sin ^2 \alpha=\frac{1}{2}=\sin ^2 \frac{\pi}{4} \\ & \Rightarrow \alpha=\frac{\pi}{4}\end{aligned}$
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