Solving set of equations having replacements

Question

I am trying to solve a coordinate transformation problem which can be reduced to determining six parameters x1, x2...x6 in terms of two variable (phi and theta) satisfying the eleven equations (all might not be independent) having few replacements. I tried to use the function "Reduce", but not getting any output at all. Will appreciate any help. Thanks

Reduce[x1*x4 + x2*x5 + x3*x6 == 0 && 
        x1*Sin[theta]*Cos[phi] + x2*Sin[theta]*Sin[phi] + 
          x3*Cos[phi] == 0 && 
        x4*Sin[theta]*Cos[phi] + x5*Sin[theta]*Sin[phi] + 
          x6*Cos[phi] == 0 && x1*x1 + x2*x2 + x3*x3 == 1 && 
        x4*x4 + x5*x5 + x6*x6 == 1  && 
        x1 /. {theta -> 0, phi -> 0} == 1 && 
        x2 /. {theta -> 0, phi -> 0} == 0 && 
       x3 /. {theta -> 0, phi -> 0} == 0 && 
      x4 /. {theta -> 0, phi -> 0} == 0 && 
     x5 /. {theta -> 0, phi -> 0} == 1 && x6 /. {theta -> 0, 
    phi -> 0} == 0, {x1, x2, x3, x4, x5, x6}]

Answer 1

If I correctly understand it,

sol=Reduce[x1*x4 + x2*x5 + x3*x6 == 0 &&  x1*Sin[theta]*Cos[phi] + x2*Sin[theta]*Sin[phi] + 
x3*Cos[phi] == 0 &&  x4*Sin[theta]*Cos[phi] + x5*Sin[theta]*Sin[phi] + x6*Cos[phi] == 0 && 
x1*x1 + x2*x2 + x3*x3 == 1 && x4*x4 + x5*x5 + x6*x6 == 1, {x1, x2, x3, x4, x5, x6}]

does the job. It takes some time and the output is too long to be citted here.

Addition. If I correctly understand your point, then the output of (23 is found by trials)

Table[sol[[j]] /. {phi -> 0, theta -> 0}, {j, 1, 23}]

{False, False, False, False, False, False, False, False, False, (x1 == -1 || x1 == 1) && x2 == 0 && x3 == 0 && x4 == 0 && (x5 == -1 || x5 == 1) && x6 == 0, (x2 == -Sqrt[1 - x1^2] || x2 == Sqrt[1 - x1^2]) && x3 == 0 && (x4 == -Sqrt[1 - x1^2] || x4 == Sqrt[1 - x1^2]) && -1 + x1^2 != 0 && x5 == (x1 x2 x4)/(-1 + x1^2) && x6 == 0, False, False, False, False, False, False, False, False, False, False, False, False}

shows the tenth solution is it. That solution can be displayed by

sol[[10]]

Sin[theta] == 0 && (x1 == -1 || x1 == 1) && x2 == 0 && x3 == 0 && x4 == 0 && (x5 == -1 || x5 == 1) && x6 == 0 && Cos[phi] != 0

Addition 2.The following answers the latest explanation of her/his question by OP.

sol = Solve[x1[phi, theta]*x4[phi, theta] + x2[phi, theta]*x5[phi, theta] + 
 x3[phi, theta]*x6[phi, theta] == 0 &&  x1[phi, theta]*Sin[theta]*Cos[phi] + 
 x2[phi, theta]*Sin[theta]*Sin[phi] + x3[phi, theta]*Cos[phi] == 
0 && x4[phi, theta]*Sin[theta]*Cos[phi] + 
 x5[phi, theta]*Sin[theta]*Sin[phi] + x6[phi, theta]*Cos[phi] == 
0 && x1[phi, theta]*x1[phi, theta] + 
 x2[phi, theta]*x2[phi, theta] + x3[phi, theta]*x3[phi, theta] == 
1 && x4[phi, theta]*x4[phi, theta] + 
 x5[phi, theta]*x5[phi, theta] + x6[phi, theta]*x6[phi, theta] == 
1 , {x1[phi, theta], x2[phi, theta], x3[phi, theta],x4[phi, theta], x5[phi, theta], x6[phi, theta]}]

This produces a very long output, fo example,

LeafCount[First[sol]]

8031

and a warning.