A line makes angles `alpha,beta,gammaa n ddelta` with the diagonals of a cube. Show that `cos^2alpha+cos^2beta+cos^2gamma+cos^2delta=4//3.`

Explanation

The four diagonals of a cube are AL, BM, CN and OP.
`" "` Direction cosines of OP are `(1)/(sqrt(3)), (1)/(sqrt(3))and (1)/(sqrt(3)).`
`" "` Direction cosines of AL are `(-1)/(sqrt(3)), (1)/(sqrt(3))and (1)/(sqrt(3))`.
`" "` Direction cosines of BM are `(1)/(sqrt(3)), (-1)/(sqrt(3)) and (1)/(sqrt(3))`.
Direction cosines of CN are `(1)/(sqrt(3)), (1)/(sqrt(3)) and (-1)/(sqrt(3))`.

Let `l, m and n` be the direction cosines of a line which is inclined at angles `alpha, beta, gamma and delta` respectively, to the four diagonals , then
`" "cosalpha=l*(1)/(sqrt(3))+m*(1)/(sqrt(3))+n*(1)/(sqrt(3))`
`" "=(l+m+n)/(sqrt(3))`
Similarly, `" "cosbeta=(-l+m+n)/(sqrt(3))`
`" "cosgamma=(1-m+n)/(sqrt(3))`
`" "cosdelta=(l+m-n)/(sqrt(3))`
`" "cos^(2)alpha+cos^(2)beta+cos^(2)gamma+cos^(2)delta=(1)/(3)[(l+m+n)^(2)+(-l+m+n)^(2)+(l-m+n)^(2)+(l+m-n)^(2)]`
`" "=(1)/(3)*4(l^(2)+m^(2)+n^(2))=(4)/(3)`