eqs = {Tan[310 Degree] == k,
      (Sin[320 Degree]-Cos[310 Degree])/(Tan[140 Degree] + Cot[220 Degree])==a}

eqs2 = eqs // TrigToExp

(*   {(I (E^(-40 I [Degree]) + E^(40 I [Degree])))/(
       E^(-40 I [Degree]) - E^(40 I [Degree])) == k,
-((I (E^(-40 I [Degree]) - E^(40 I [Degree])))/(-((
 I (E^(-40 I [Degree]) - E^(40 I [Degree])))/(
 E^(-40 I [Degree]) + E^(40 I [Degree]))) - (
I (E^(-40 I [Degree]) + E^(40 I [Degree])))/(
E^(-40 I [Degree]) - E^(40 I [Degree])))) == a}   *)

eli = Eliminate[
        eqs2 /. (E^(-40 I [Degree]) -> 1/c) /. (E^(40 I [Degree]) ->c), c]

(*    (-4 - a^2) k^2 - a^2 k^4 + a^2 k^6 == -a^2   *)

sol=Solve[eli, a]

(*   {{a -> -((2 k)/Sqrt[1 - k^2 - k^4 + k^6])},
      {a -> (2 k)/Sqrt[1 - k^2 - k^4 + k^6]}}      *)

eqs /. First@sol // Reduce

(*   False   *)

eqs /. Last@sol // Reduce

(*   k == -Cot[40 Degree]   *)