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If `overset(to)(u) , overset(to)(v) , overset(to)(w)` are three non- coplanar unit vectors and `alpha , beta , gamma` the angles between `overset(to)(u) " and " overset(to)(v) , overset(to)(v) " and " overset(to)(w), overset(to)(w)" and " overset(to)(u)` respectively and `overset(to)(x) , overset(to)(y) , overset(to)(z) ` are unit vectors along the bisectors of the angles `alpha , beta , gamma` respectively . Prove that
`[overset(to)(x) xx overset(to)(y) overset(to)(y) xx overset(to)(z) overset(to)(z) xx overset(to)(x)] = (1)/(16) [ overset(to)(u) overset(to)(v) overset(to)(w) ]^(2) " sec"^(2) . (alpha)/(2) " sec"^(2) .(beta)/(2) " sec"^(2) . (gamma)/(2)`

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Since `[vec(x) xx vec(y) vec(y) xx vec(z) vec(z) xx vec(x)] =[vec(x) vec(y) vec(z)]^(2)` [from Eq .(i)]
`=(1)/(64) sec^(2) .(a)/(2) . Sec^(2) . (beta)/(2) sec^(2) . (gamma)/(2) [vec(u)+vec(v)vec(v)+vec(w)vec(w)+vec(u)]^(2)……(i)`
and `[ve(u)+vec(v)vec(v)+vec(w)vec(w)+vec(u)]=2[vec(u)vec(v)vec(w)].....(iii)`
`:. [vec(x) xx vec(y) vec(y) xx vec(z) vec(z) xx vec(x)]`
`=(1)/(64) sec^(2) . (a)/(2). sec^(2) .(beta)/(2) . sec^(2).(gamma)/(2).4 [vec(u)vec(v)vec(w)]^(2)`
`=(1)/(16). [vec(u)vec(v)vec(w)]^(2)sec^(2).(a)/(2).sec^(2).(beta)/(2).sec^(2).(gamma)/(2)`
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