Clarification and Attempted Solution
I've been stuck on this problem for several days now, and I'm entirely frustrated at this point. I do not know how to estimate the integral in the particular way needed to obtain the desired inequalities. I have tried playing around with $$ \int_Xc_k\cos^{\sqrt{2k}}\sin t\ dt $$ but to no avail, and I am aware of the identity $$ \frac{1+\cos t}{2}=\cos^2{t/2}, $$ but again I cannot see how any of this can help me come to a solution. Apologies if it seems I have not put much effort into this problem, but I just have 2-3 pages of seemingly useless inequalities written down which I see no reason to post here.
Note
I do not want a full solution, just a hint to get me going in the right direction. Leave me as much work as possible!
Problem Statement:
The constants $c_k$ are such that $k^{-1}c_k$ is bounded. Estimate the relevant integral more precisely and show that $$ 0<\lim_{k\rightarrow\infty}k^{-1/2}c_k<\infty. $$
Relevant Information:
The 'relevant integral' is the Riemann integral $$ \frac{c_k}{\pi}\int_0^\pi\left(\frac{1+\cos t}{2}\right)^k\ dt, $$ and the estimate mentioned in the problem is $$ 1=\frac{c_k}{\pi}\int_0^\pi\left(\frac{1+\cos t}{2}\right)^k\ dt>\frac{c_k}{\pi}\int_0^\pi\left(\frac{1+\cos t}{2}\right)^k\sin t\ dt=\frac{2c_k}{\pi(k+1)}. $$
The constants $c_k,\ k\in\mathbb{Z}^+$ are defined by the leftmost equality above. The relevant pages are in Section 4.24 of Rudin's Real and Complex Analysis.