Let O, one vertex of a cube be the origin and three edges through O along the co-ordinate axes. The four diagonals are OP,AA′,BB′ and CC′. Let ' a′ be the length of each edge. Then the coordinates of P,A,A′ are (a,a,a),(a,0,0),(0,a,a) The direction ratios of OP are a,a,a. The direction cosines of OP are
a
a√3
,
a
a√3
,
a
a√3
i.e.,
1
√3
,
1
√3
,
1
√3
Similarly, direction cosines of AA′ are (−
1
√3
,
1
√3
,
1
√3
) Let θ be the angle between the diagonals OP and AA′, then cos‌θ=
1
√3
(−
1
√3
)+
1
√3
(
1
√3
)+
1
√3
(
1
√3
) =−
1
3
+
1
3
+
1
3
=
1
3
⇒cos‌θ=
1
3
⇒θ=cos−1(
1
3
) Thus, the angle between any two diagonals of a cube is cos−1(