#33 Measure Your Calibre

Relevant wiki: Euler's Formula

$\begin{aligned} S & = \sum_{r=0}^n {n \choose r} \cos r\theta \\ & = \sum_{r=0}^n {n \choose r} \Re \left \{e^{ir\theta} \right \} & \small \color{#3D99F6} \text{By Euler's formula: } e^{ix} = \cos x + i \sin x \\ & = \Re \left \{\sum_{r=0}^n {n \choose r} e^{ir\theta} \right \} & \small \color{#3D99F6} \text{where }\Re\{z\} \text{ is the real part of complex number } z. \\ & = \Re \left \{\left(1+e^{i\theta} \right)^n \right \} \\ & = \Re \left \{\left(1+\cos \theta + i\sin \theta \right)^n \right \} \\ & = \Re \left \{\left(1+2\cos^2 \frac \theta 2 - 1 + 2 i \sin \frac \theta 2 \cos \frac \theta 2 \right)^n \right \} \\ & = \Re \left \{2^n \cos^n \frac \theta 2 \left( \cos \frac \theta 2 + i \sin \frac \theta 2 \right)^n \right \} \\ & = \Re \left \{2^n \cos^n \frac \theta 2 \cdot e^{i \frac {n\theta} 2} \right \} \\ & = \boxed {2^n \cos^n \dfrac \theta 2 \cos \dfrac {n\theta} 2} \end{aligned}$

$\large{\displaystyle{\sum^n_{r=0}} \binom{n}{r} \cos r \theta }$ $\implies$

LET ME THINK ABOUT THE BONUS!