ClearAll["Global`*"];
y[t_] := r[t] Cos[ω t] + l Sin[ω t];
x[t_] := l Cos[ω t] - r[t] Sin[ω t];
y'[t]

x'[t]

FullSimplify[x'[t]^2 + y'[t]^2]

ClearAll["Global`*"];
xx[t_] := (l+x[t])Sin[θ[t]];
yy[t_] := -(l+x[t])Cos[θ[t]];
xx'[t]

yy'[t]

FullSimplify[xx'[t]^2 + yy'[t]^2]