Weierstrasssubstitution `\[Psi] -> 2 ArcTan[u\[Psi]]`  is the way to solve these equations

    eqn = {-a r Cos[\[Phi]] == R Cos[\[Psi]], 
    r Sin[\[Phi]] == R Sin[\[Psi]]} /. \[Psi] -> 2 ArcTan[u\[Psi]] //TrigExpand

    solu\[Psi] = Solve[eqn, {R, u\[Psi]}]  ;
    sol\[Psi] = solu\[Psi] /. u\[Psi] -> Tan[\[Psi]/2]
    (*{{R -> -r Sqrt[a^2 Cos[\[Phi]]^2 + Sin[\[Phi]]^2],
    Tan[\[Psi]/2] ->a Cot[\[Phi]] -Csc[\[Phi]] Sqrt[a^2 Cos[\[Phi]]^2 +Sin[\[Phi]]^2]}
    , 
    {R ->r Sqrt[a^2 Cos[\[Phi]]^2 + Sin[\[Phi]]^2],
    Tan[\[Psi]/2] ->Csc[\[Phi]] (a Cos[\[Phi]] + Sqrt[a^2 Cos[\[Phi]]^2 +Sin[\[Phi]]^2])}}*)

Knowing  

    Tan[x] == (2 Tan[x/2])/(1 - Tan[x/2]^2) // FullSimplify (*True*)

we get 

    tan\[Psi] = (2 Tan[ \[Psi]/2])/(1 - Tan[ \[Psi]/2]^2) /. sol\[Psi]// FullSimplify
    (* {-(Tan[\[Phi]]/a), -(Tan[\[Phi]]/a)}*)

Hope it helps!