Chebyshev Polynomial Practice: Sum

Let $T_n$ be the nth Chebyshev Polynomial where $T_n(\cos(x))=\cos(nx)$

We have $\displaystyle \sum_1^{45}\cos^2 (2k-1)=\dfrac{\displaystyle \sum_1^{90}\cos^2 (2k-1)}{2}$

Now we are looking at the sum of the roots of the 90th Chebyshev Polynomial. Note that since we are working with $cos^{\textbf{2}}$ here we have $T_{90}(\sqrt{x})$ which still has 45 roots. This will still be a polynomial because we have $T_{even}$ , so all $x$ will have an even power.

The lead coefficient of the nth polynomial will be $2^{n-1}$ and the second coefficient will be $-2^{n-2}\cdot\frac{n}{2}$

Thus by vietas we have

$-\dfrac{-2^{43}\cdot45}{2^{44}}=\boxed{22.5}$