`vecu, vecv and vecw` are three nono-coplanar unit vectors and `alpha, beta and gamma` are the angles between `vecu and vecu, vecv and vecw and vecw and vecu`, respectively and `vecx , vecy and vecz` are unit vectors along the bisectors of the angles `alpha, beta and gamma.` respectively, prove that `[vecx xx vecy vecy xx vecz vecz xx vecx) = 1/16 [ vecu vecv vecw]^(2) sec^(2) alpha/2 sec^(2) beta/2 sec^(2) gamma/2`.

Explanation

WE know that `[vecx xx vecy vecy xx veczveczxx vecx] = [vecx vecy vecz] ^(2)` Also a vector along the bisector of given two unit vectors
`vecu,vecv,"is " vecu +vecv`
A unit vector along the bisector is `(vecu +vecv)/(|vecu +vecv|)`
`|vecu +vecv|^(2)=1+1+2vecu.vecv=2+2cosalpha=4cos^(2)""alpha/2`
`Rightarrow vecx = (vecu +vecv)/(2cos ""alpha/2)`
similarly , ` vecy= (vecv + vecw)/(2cosbeta//2) and vecz= (vecu+vecw)/(2cosgamma//2)`
`Rightarrow [ vecxvecy vecz] = 1/8[vecu+vecv vecv + vecw vecu + vecw] xx sec "" alpha/2 sec"" beta/2 sec"" gamma/2`
`= 1/82[vecu vecv vecw] sec""alpha/2sec"" beta/2sec"" gamma/2`
`1/4[vecu vecv vecw] sec""alpha/2 sec""beta/2sec""gamma/2`
`Rightarrow [vecx xx vecyvecy xxvecz vecz xx vecx] = [vecx " "vecy " " vecz]^(2)`
`= 1/16[vecu vecv vecw]^(2) sec^(2)""alpha/2 sec^(2)""beta/2 sec^(2)""gamma/2`