The function $f(x)=\cos x-2px$ is monotonically decreasing for:

A function is monotonically decreasing if $f^{\prime }(x)\le 0$ for all $x$.
$$f(x)=\cos x-2px$$
$$f^{\prime }(x)=-\sin x-2p\le 0$$
$$-\sin x\le 2p⟹\sin x+2p\ge 0$$
$$2p\ge -\sin x$$
The maximum value of $-\sin x$ is $1$. For the inequality to hold for all $x$:
$$2p>1⟹p>1/2$$