Hi Sander,
thank you for your reply.
I define my equations as (following there is the first one, but the other 5 equations are of the same form):
eq1[x1, x2, x3, x4, x5, x6] := x2 (Cos[x5] Cos[x3 x6] - Cos[x4] Sin[x5] Sin[x3 x6]) + x1 Sin[x4] Sin[x5];
I solve the system of equations with Reduce as follows:
sol = Reduce[
eq1[x1, x2, x3, x4, x5, x6] == eq1[x10, x20, x30, 0, 0, 0] &&
eq2[x1, x2, x3, x4, x5, x6] == eq2[x10, x20, x30, 0, 0, 0] &&
eq3[x1, x2, x3, x4, x5, x6] == eq3[x10, x20, x30, 0, 0, 0] &&
eq4[x1, x2, x3, x4, x5, x6] == eq4[x10, x20, x30, 0, 0, 0] &&
eq5[x1, x2, x3, x4, x5, x6] == eq5[x10, x20, x30, 0, 0, 0] &&
eq6[x1, x2, x3, x4, x5, x6] == 0, {x1, x2, x3, x4, x5, x6}]
And after that I use FullSimplify with the assumptions for my problem:
FullSimplify[sol,
x10 [Element] Reals && x20 [Element] Reals && x20 > 0 &&
x30 [Element] Reals && x30 > 0 && x6 [Element] Reals &&
x1 [Element] Reals && x2 [Element] Reals && x2 > 0 &&
x3 [Element] Reals && x3 >= 0 && x5 [Element] Reals &&
x5 >= 0 && x5 < 2 Pi && x4 [Element] Reals && x4 >= 0 && x4 < Pi]
The solution I get contains several logical expressions and one of the solutions is the following:
x1 == x10 && (C[1] | C[2] | C[3]) [Element] Integers &&
x4 == 2 [Pi] C[3] && x3 == x30 && [Pi] + 2 [Pi] C[1] == x4 &&
x2 == x20 && [Pi] + 2 [Pi] C[2] == x30 x6