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Question

A line makes angles α,β,γ and δ with the diagonals of a cube. Then cos2α+cos2β+cos2γ+cos2δ=x3. Find x.

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Solution

cosα=l+m+n3

cosβ=l+mn3

cosγ=lm+n3

cosδ=l+m+n3 [L2+M2+N21]

cos2α+cos2β+cos2γ+cos2γ

=13+(l+m+n)2+(l+mn)2+(lm+n)2+(l+mn)

=13(4(l2+m2+n2))=x3

43=x3(x=4)


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