Bounding $|\cos (2n+1)\theta |$ I am trying to find a bound for $|\cos (2n+1)\theta |$ by something of the form $c(n)|\cos \theta|$. I am suspecting that we can bound it by $(2n+1)|\cos \theta|$ but I'm not sure how to show that this is true. 

Use $2\cos t=e^{it}+e^{-it}$.
Then
$$\frac{\cos(2n+1)t}{\cos t}=\frac{e^{(2n+1)it}+e^{-(2n+1)it}}{e^{it}+e^{-it}}=e^{2nit}-e^{2(n-1)t}+e^{2(n-2)it}-\cdots+e^{-2nit}.$$
As each of these exponentials is bounded by $1$ is absolute value,
you get
$$|\cos(2n+1)t|\le(2n+1)|\cos t|.$$
This is the best possible; take $t$ close to $\pi/2$, then $e^{-2it}$ is close to $-1$ etc.
