We recall that discrete Cosine transform of type 5 and discrete Sine transform of type 8 are given as follows: $$C_5=\left(\cos kl\frac{\pi}{n-\frac{1}{2}}\right)_{0\leq k,l\leq n-1}~,~S_8=\left(\sin (k+\frac12)(l+\frac12)\frac{\pi}{n-\frac{1}{2}}\right)_{0\leq k,l\leq n-1}$$
Any suggestion to compute the trace of these two transforms?
$$\begin{align} &\sum_{k=0}^{n-1} \cos\frac{4k^2\pi}{2n-1} \tag1\\[6pt] &\sum_{k=0}^{n-1} \sin\frac{(2k+1)^2\pi}{2(2n-1)} \tag2 \end{align}$$