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  4. If a line in the space makes angles α, β and γ with the coordinate axes, then cos 2 α+ cos 2 β+ cos 2 γ+ sin 2 α+ sin 2 β+ sin 2 γ=

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Q. If a line in the space makes angles $\alpha, \beta$ and $\gamma$ with the coordinate axes, then $\cos 2 \alpha+\cos 2 \beta+\cos 2 \gamma+\sin ^{2} \alpha+\sin ^{2} \beta+\sin ^{2} \gamma=$

Solution:

$\cos 2 \alpha+\cos 2 \beta+\cos 2 \gamma+\sin ^{2} \alpha+\sin ^{2} \beta+\sin ^{2} \gamma$
$=2 \cos ^{2} \alpha-1+2 \cos ^{2} \beta-1+2 \cos ^{2} \gamma-1+1-\cos ^{2} \alpha+1-\cos ^{2} \beta+1-\cos ^{2} \gamma$
$=\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1$