Please someone help me to make sure my prove that Sn bounded is correct Please someone help me to make sure my prove that $S_n$ bounded is correct.

I want to prove that $S_n=sum \sin(2n+1)x$ is bounded. I am sure about $$S_n=\frac{\cos(0)x-\cos(2n+2)x}{2\sin x}$$ $$|S_n|\le\frac{2}{2|\sin x|}$$ $$|S_n|\le\frac{2}{2|\sin x|}=\frac{1}{|\sin x|}.$$

But I am not sure about  $|\cos(0)x-\cos(2n+2)x|\le2.$
For $\lvert \cos(0)x−\cos(2n+2)x\rvert\leq 2$.

The maximum value of a cosine is $1$, so the difference between two cosines cannot be greater than $2$
For $\lvert \cos(0)x−\cos(2n+2)x\rvert\leq 2$.

The maximum value of a cosine is $1$, so the difference between two cosines cannot be greater than $2$