It was shown in here that $$\left\lvert \frac{\sin(nx)}{n\sin(x)}\right\rvert \le1\,\,\forall x\in\mathbb{R}-\{\pi k: k\in\mathbb{Z}\}$$ iff $n$ is a non-zero integer.
Using the similar argument in the same post, we are able to show that $$\left\lvert \frac{\cos(nx)}{n\cos(x)}\right\rvert \le1\,\,\forall x\in\mathbb{R}-\{\frac{(2k+1)\pi}{2} : k\in\mathbb{Z}\}$$ iff $n$ is an odd non-zero integer (Please alert me if this is wrong).
Now, by some graph sketching, it seems that $$\left\lvert \frac{\sin(nx)}{n\sin(x)} + \frac{\cos(nx)}{n\cos(x)} \right\rvert \le\left\lvert \frac{n+1}{n}\right\rvert \,\,\forall x\in\mathbb{R}-\{\frac{k\pi}{2} : k\in\mathbb{Z}\}$$ iff $n$ is an odd non-zero integer.
I am not sure if the above inquality is true. Please enlighten me!