# Tagged Questions

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### Using the formulas de Moivre to deduce trigonometric identities.

Yesterday I made a test of complex variables, and this contained a question (in which I could not solve) that asked to use the de Moivre formulas to deduce the following trigonometric identities: ...
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### Compute the Integral

Compute the integral. $$\int_{-\infty}^\infty \frac{x^4}{1+x^8} \, dx$$ The answer at the back of the book is $$\frac{\pi}{4\sin(\frac{3\pi}{8})}$$
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### Is the complex cosine function surjective?

Let $\cos z=\frac{e^{iz} - e^{-iz}}{2}$ be the complex cosine function. Then is $\cos:\mathbb{C}\rightarrow \mathbb{C}$ surjective? If so, how do i prove this?
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### Product of $1-\operatorname{cis}(2k\pi/n)$

I'm in a question about polygonals and got stuck at a part. I have to prove that $$\prod_{k=1}^{n-1} \left(1 - \operatorname{cis}(\frac{2k\pi}{n})\right) = n$$ I've tried to multiply it to make ...
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### Complex and Trigonometric Identities

How can I get this result: $$\frac{1+cis\theta}{1-cis\theta}=-\frac{1}{i\tan(\theta/2)}$$ I've tried to expand $1-cis\theta$ as $(1+cis(\theta/2))(1-cis(\theta/2))$, but it doesn't help.
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### Can you get the exact real value of $\left((-1)^{\frac{1}{180}}\right)^{89}-\left((-1)^{\frac{1}{180}}\right)^{91}$?

By using euler formula,one can obtain: $$2\sin\left(\frac{\pi}{180}\right)=\left((-1)^{\frac{1}{180}}\right)^{89}-\left((-1)^{\frac{1}{180}}\right)^{91}.$$ In order to get the exact real value of ...
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### Trigonometric manipulation of complex number, how does this step occur?

I was reading the section about DeMoivre, and my book showed how to derive his formulas. The next part is supposed to be about finding roots of complex and real numbers. Roughly, it says: "Let $z$ be ...
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### Prove a Trigonometric Series

Question: $$\cot^{2}\frac{\pi }{2m+1}+\cot^{2}\frac{2\pi }{2m+1}+\cdots+\cot^{2}\frac{m\pi }{2m+1}=\frac{m(2m-1)}{3}$$ $m$ is a positive integer. Attempt: I started by showing that ...
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### When are we (not) allowed to replace $x$ by $ix$?
It seems to be quite a common manipulation to replace $x$ by $ix$. Every time I see it's being done in a textbook, I blindly trust the author without really understanding when are we allowed to do so ...
Can any ne help me to prove this problem? $x$ is called rational angle if $x=a\pi$ for $a\in \mathbb{Q}$. Let $0<x<\pi/4$ be a rational angle, prove that $\tan x$ is irrational. Let ...