I think your result has a typo. Instead of -ET (2*m*n + B*R0^2)/(4*Pi*R0)
it should read -ET (-2*m*n + B*R0^2)/(4*Pi*R0)
, otherwise there is no solution.
A further problem is the minus sign in front of the Cos term. MMA always uses the full form of an expression for matching. Look at:
((-2 m n + B R0^2) Cos[t])/(4 \[Pi] R0) // FullForm
(* Times[Rational[1,4],Power[Pi,-1],Power[R0,-1],Plus[Times[-2,m,n],Times[B,Power[R0,2]]],Cos[t]] *)
and
-(((-2 m n + B R0^2) Cos[t])/(4 \[Pi] R0)) // FullForm
(* Times[Rational[-1,4],Power[Pi,-1],Power[R0,-1],Plus[Times[-2,m,n],Times[B,Power[R0,2]]],Cos[t]] *)
You see, the minus is integrated in the Rational[-1,4]. Therefore, for matching, the minus can not be matched separately. Here is how you can do it:
{((-2*m*n + B*R0^2)*Sin[t])/(4*Pi*R0), -1/4*((-2*m*n + B*R0^2)*Cos[t])/(Pi*R0), 0} /. { x__ Sin[t], y__ Cos[t], 0} :> ET y
