<s>[INST] Compute $\sum_{k=0}^{180} \cos^22k^\circ $ I'm trying to compute the following:
$$\sum_{k=0}^{180} \cos^22k^\circ $$
 [/INST] Remark that:
$cos(x+90°) = sin(x)$
And that:
$cos^2(x) + sin^2(x) = 1$
So you can group your terms:
$$
\begin{align*}
\sum_{x=0}^{180} cos^2(x.2°) &=  \sum_{x=0}^{44} cos^2(x.2°) \\
                             & + \sum_{x=45}^{89} cos^2(x.2°) \\
                             & + \sum_{x=90}^{134} cos^2(x.2°) \\
                             & + \sum_{x=135}^{179}{cos^2(x.2°)} \\
                             & + cos^2(180.2°) \\
\\
&= \sum_{x=0}^{44}{cos^2(x.2°)+cos^2(x.2°+90°)} \\
& + \sum_{x=90}^{179}{cos^2(x.2°)+cos^2(x.2°+90°)} \\
& + cos^2(360°) \\
\\
&= \sum_{x=0}^{44}{cos^2(x.2°)+sin^2(x.2°)} \\
& + \sum_{x=90}^{134}{cos^2(x.2°)+sin^2(x.2°)} \\
& + 1 \\
\\
&= \sum_{x=0}^{44}{1} \\
& + \sum_{x=90}^{134}{1} \\
& + 1 \\
&= 91
\end{align*}
$$
See the approximate results at wolfram alpha
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