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If a line makes $30^{\circ}$ with the positive direction of $\mathrm{x}$ -axis, angle $\beta$ with the positive direction of $y$ -axis and angle $\gamma$ with the positive direction of $\mathrm{z}$ -axis, then what is $\cos ^{2} \beta+\cos ^{2} \gamma$ equal to ?
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The correct answer is:
$1 / 4$
Direction cosines are $\cos 30^{\circ}, \cos \beta$ and $\cos \gamma$
Since we know $\cos ^{2} 30+\cos ^{2} \beta+\cos ^{2} \gamma=1$
$\Rightarrow \cos ^{2} \beta+\cos ^{2} \gamma=\frac{1}{4} \quad\left(\because \cos 30^{\circ}=\frac{\sqrt{3}}{2}\right)$
Since we know $\cos ^{2} 30+\cos ^{2} \beta+\cos ^{2} \gamma=1$
$\Rightarrow \cos ^{2} \beta+\cos ^{2} \gamma=\frac{1}{4} \quad\left(\because \cos 30^{\circ}=\frac{\sqrt{3}}{2}\right)$
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