How about the following?
(*Define the two functions. *)
f[x_] := Cos[x]^2; g[x_] := 1/2;
(*Find the intercepts and thread them.*)
xv = x /. Solve[f[x] == g[x], x] ; yv = f[xv]; pts = Thread[{xv, yv}]
(* Overlay the original and intercept plots using Show[].*)
Show[
(*Plot original functions*)Plot[{f[x], g[x]}, {x, -2 Pi, 2 Pi}(*End Plot*)],
(*Plot the intercepts*) ListPlot[pts, PlotMarkers -> Style[\[FilledCircle],18](*End ListPlot*)](*End Show*)]