Finding a closed form for $\cos{x}+\cos{3x}+\cos{5x}+\cdots+\cos{(2n-1)x}$

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  • $\sum \cos$ when angles are in arithmetic progression [duplicate]

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Let $z=\cos\theta+i\sin\theta$ i.e. $z=e^{i\theta}$

Your sum:$$e^{i\theta}+e^{3i\theta}+e^{5i\theta}+…e^{(2n-1)i\theta}$$

This is a GP with common ratio $e^{2i\theta}$

Therefore sum is $$\frac{a(r^n-1)}{r-1}$$
$$\frac{e^{i\theta}(e^{2ni\theta}-1)}{e^{2i\theta}-1}$$
$$\frac{(\cos \theta+i\sin\theta)(\cos(2n\theta)+i\sin\theta-1)}{\cos(2\theta)+i\sin(2\theta)-1}$$

Computing it’s real part should give you the answer

Acknowledgement:Due credits to @LordShark Idea

Following @TheDeadLegend’s answer I found this telescoping technique. Turns out that you need a similar identity to make it work:

$$ \sin(\alpha + \beta) – \sin(\alpha – \beta) =2\cos \alpha \sin \beta$$


\begin{align}
g(x) &= \sum_{k=1}^n \cos(2k-1)x \\
&= \frac{1}{2\sin x}\sum_{k=1}^n 2\cos(2k-1)x \cdot \sin x \\
&= \frac{1}{2\sin x}\sum_{k=1}^n \left[\sin 2kx – \sin2(k-1)x \right] \\
&= \frac{1}{2\sin x}(\sin 2nx-0) = \frac{\sin 2nx}{2\sin x}.
\end{align}