ClearAll["Global`*"];
x[t_] := R Cos[ω t] + l Sin[ϕ[t]];
y[t_] := R Sin[ω t] - l Cos[ϕ[t]];
x'[t]^2 + y'[t]^2

FullSimplify[%]

ClearAll["Global`*"];
g = 9.8; l = 5.0; R = 1.0; ω = 1.0; tmax = 50;
soln = NDSolveValue[{ϕ''[t]+(g/l)Sin[ϕ[t]] == (R/l)ω^2 Cos[ϕ[t]-ω t], ϕ'[0]==0, ϕ[0]==0}, ϕ[t], {t, 0, tmax}] ;
Plot[soln, {t, 0, tmax}]

ClearAll["Global`*"];
x[t_] := R Cos[ϕ[t]+ω t] + R Cos[ω t];
y[t_] := R Sin[ϕ[t] + ω t] + R Sin[ω t];
x'[t]^2 + y'[t]^2

FullSimplify[%]

Expand[%]