# Tagged Questions

For questions about or related to trigonometric series.

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### Closed form for $\prod_{l=1}^\infty \cos\dfrac{x}{3^l}$

Is there any closed form for the infinite product $\prod_{l=1}^\infty \cos\dfrac{x}{3^l}$? I think it is convergent for any $x\in\mathbb{R}$. I think there might be one because there is a closed form ...
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### Explaining a Series Expansion for the Divisor Function

The "sum-of-divisors" function, defined as $\sigma(n)=\sum_{d\,\mid\,n}d$, can be expressed as $$\sigma(n)=\sum_{i=1}^n\sum_{j=1}^i\cos\Big(2\pi n \frac{j}{i}\Big)$$This expansion makes logical sense. ...
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### Sum of cosines with a multiplicative factor in the angle and different interval

I have found the following formula for the sum of cosines in both here and here. \begin{align} \sum^n_{l=1} \cos \left(\frac{2 \pi l}{n}\right) = 0 \end{align} I would like to know what the sum ...
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### a Proof onInverse Trigonometry

Ff $\arcsin x + \arcsin z + \arcsin z = 1.5\pi$, Prove that $x^{2006}+y^{2007}+z^{2008}-\frac 9{x^{2006}+y^{2007}+z^{2008}}=0$.
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### How does $\cos(x)\cdot\cos\left(\frac{3}{2}x\right)$ become $\frac{1}{2}\left(\cos\left(\frac{1}{2}x\right) + \cos\left(\frac{5}{2}x\right)\right)$?

How can you rewrite $\cos(x)\cdot\cos\left(\frac{3}{2}x\right)$ to $\frac{1}{2}\left(\cos\left(\frac{1}{2}x\right) + \cos\left(\frac{5}{2}x\right)\right)$? What rules have been used? I need it on ...
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### Prove $a_n = \sin(n\pi /2)$ does not converge to $0$

Not sure how to prove it doesn't converge to $0$. Prove by contradiction? How do I assume that the sequence converges to $0$ since if I do the partial sums it does end up adding to $0$.
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### How to find the scaling with N for FWHM and MAX of $F(X) = (1+\cos((2N+1)\pi X)/(1+\cos(\pi X))$?

Given a function $F(X) = (1+\cos((2N+1)\pi X)/(1+\cos(\pi X))$. Peaks are at odd $X$ integer values. How to find the scaling with $N$ of the FWHM (full width half max) and peak max? For example for ...
### How to find $\sum_{k=1}^nk\sin\left(k\pi/n\right)$ [duplicate]
I would appreciate if somebody could help me with the following problem Q: How to find ? $$\sin\left( \frac{\pi}{n}\right)+2\sin\left( \frac{2\pi}{n}\right)+\cdots+ n\sin\left( \frac{n\pi}{n}\right)$$...