Question: Elimination of variables from set of equations

Hello,

I have a problem with the elimination of variables from set of eq.

For example:

I tried:

>  eliminate({x=cos(v) *cos(u), y=cos(v)*sin(u), z=sin(v)},{u,v});
                                   y
  [{v = arcsin(z), u = arcsin(-----------)},
                                    2 1/2
                              (1 - z )

                         /      2    2\1/2
                   2 1/2 |-1 + z  + y |
        {x - (1 - z )    |------------|   }]
                         |        2   |
                         \  -1 + z    /

but I need the result like  x^2+y^2+z^2-1=0

Other example:

I tried eliminate({x=cos(v), y=sin(v)},{v}) in Mathematica 6 and I got  x^2+y^2=1,

If I try the same in Maple, I get

[{v = arcsin(y)}, {x-(1-y^2)^(1/2)}]

Could you help me please?

Thank you.

Vladimir