How can I get right solution in sagemath?

Question

I want to find the intersection of the cylinders given parametrically:

FC1 = vector([cos(alpha), sin(alpha), z])

FC2 = vector([z, cos(alpha), sin(alpha)])

FC3 = FC2-FC1

The program gives a solution: solve([FC3[0] - x== 0,FC3[1]-y==0,FC3[2]==0],[x,y,z]) -->

[[x == cos(alpha) - sin(alpha), y == -cos(alpha) + sin(alpha), z == sin(alpha)]]

The right decision should be like that:

[cos(asin(cos(alpha))),sin(asin(cos(alpha))),sin(alpha)]