Trying to reproduce your question I get same results in both cases

Solve[Sin[2 x] == 1 + Cos[2 x], x] == Solve[Sin[2 x] - Cos[2 x] == 1, x]
(* True*)

Trying to reproduce your question I get same results in both cases

Solve[Sin[2 x] == 1 + Cos[2 x], x] == Solve[Sin[2 x] - Cos[2 x] == 1, x]
(* True*) 

{{x -> ConditionalExpression[1/2 (\[Pi]/2 + 2 \[Pi] C[1]),C[1] \[Element] Integers]},
{x ->ConditionalExpression[1/2 (\[Pi] + 2 \[Pi] C[1]),C[1] \[Element] Integers]}}

The solutions are x=Pi/2+k Pi and x=Pi/4+l Pi , k,l Integers!

as expected!