Let $n$ be a positive integer. Is $\cos (2\pi/n)$ irrational if $n$ is not $1$, $2$, $3$, $4$ or $6$?
I couldn’t find any proof on the internet. I would appreciate an elementary proof , but any help would be appreciated!  Thank you.
We have to evaluate the following integral:  $$\int \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} dx$$  I tried this:  I multiplied both the numerator and denominator by $\sec x$ And substituted $\tan x = t$.  But after that I got stuck.  The book where this is taken from gives the following as the answer: $$\ln(1+t)-\frac14\ln(1+t^4)+\frac1{2\sqrt2}\ln\frac{t^2-\sqrt2t+1}{t^2+\sqrt2t+1}-\frac12\tan^{-1}t^2+c$$ where $t=\sqrt{\cot x}$ Putting a piece of code in between two html tags in Solaris On a solaris HTML file, how do I insert between an html tag and head tag? through bash scripting.  I have an html tag that looks like this:  &lt;html xmlns="http://www.w3.org/1999/xhtml" xmlns:arf="http://scap.nist....-format/1.1" lang="en"&gt;   I want to insert a script written in text file, inside this 'SOLARIS' run HTML file, AFTER the long &lt;html....&gt; tag and BEFORE &lt;head&gt; tag, meaning BETWEEN the two tags, How do I do that?  I want to test using the bash commands to do it for one file at a time as well as multiple file in a folder directory. Will I have to use the folder path in my BASH code? Will I have to use FOR loop or interation in my BASH code? What would be the right code. Note: it is SOLARIS run html file not LINUX one. The name of my HTML file is report.usc20.html.  These are the lines of script that I want to use that are in my backbutton_and_scroller_script.txt file:  &lt;div id="button"&gt;&lt;a href="http://.....index.php"&gt;Back To The Hardening Page&lt;/a&gt;&lt;/div&gt; &lt;script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.3/jquery.min.js"&gt;&lt;/script&gt; &lt;script type="text/javascript" src="http://arrow.scrolltotop.com/arrow78.js"&gt;&lt;/script&gt; &lt;noscript&gt;Not seeing a &lt;a href="http://www.scrolltotop.com/"&gt;Scroll to Top Button&lt;/a&gt;? Go to our FAQ page for more info.&lt;/noscript&gt; Fit a table into one page Trying to dual boot Ubuntu 14.10 with Windows 10, last one not detected I have a laptop, and I'm going to install Ubuntu (I have experience, and I have  successfully installed , Ubuntu, , , , , , etc.), but installing  (Utopic Unicorn) alongside , Ubuntu can't detect Windows.  It shows me like there's no OS, but I have Windows&nbsp;10.  I have two primary partitions by the way if I manually install. If the Ubuntu installer can't detect Windows&nbsp;10, will it be detected at ? I was just looking at the statistics for migrated questions in the  mod tools, and noticed this oddity:    The actual statistics, as visible to any 10k+ SFF user , are: 6 migrated to Worldbuilding, 2 to Literature, 2 to SFF meta, 1 to Movies &amp; TV, 1 to Writers, and 1 to Aviation.  Why is one of the meta migrations showing up with a blank icon rather than with the other?  Both migrated questions were closed as duplicates on meta. Both were migrated by 5 community members without mod involvement. One of the stubs on main has since been deleted, but so have the stubs for some of the other migrated questions. I can't think of anything to explain this discrepancy. Polynomials with integer coefficients such that $p(a)=b,p(b)=c,p(c)=a$ With reference to:    I removed the Cortana packages from my running Windows 10 using the method posted by magicandre1981 and using  win6x_registry_tweak.   My question is how can I re-install Cortana (or for that matter any other package removed similarly)?  I have opened up the install.wim image (converted from a Win 10 install.esd file) and found Cortana in the system apps folder, but don't know how / what to use to re-install it.  Any help please? How to find integer solutions to $M^2=5N^2+2N+1$? What is difference between int.class and Integer.TYPE in java? 407 Proxy Authentication Required When I try to install a software using Ubuntu Software center I get:  Failed to download repository information Check Your Internet connection   When I try to do a apt-get install something, I get:  407  Proxy Authentication Required   I use a proxy server that requires a user-name and a password. I have set my systems proxy manually, by plugging in the required numbers in the Networks proxy and applied it system wide. I guess the problem now is plugging in my user-name and password.  When I use INTERNET via Mozilla, it specifically asks me for my user-name and password. I just found http://programmingfaq.w3ec.com/faq/4761/whats-the-hi-lo-algorithm which has an exact copy of this Stack Overflow question, , with all its answers and no difference in a single character. There is no reference to Stack Overflow.  Is this legal, or at least tolerated? Is this impertinent theft of information? I just can't see any sense in copying whole pages, particularly when not referencing the source. Biblatex 3.3 name formatting I would like to create a custom format for citing authors' names in the text, let's call it "newformat". One option I can think of is creating a new bibmacro for names where the parameters are changed to my liking, such as   \documentclass{article}    \usepackage{filecontents}   \usepackage[backend=biber]{biblatex}[2016/01/01] %the latest version of biblatex?   % biber is version 2.4; the latest?  \begin{filecontents}{\jobname.bib} @misc{Doe12,   author = {Doe, John},   year = {2012},   title = {A macro for formatting names}, } \end{filecontents}  \addbibresource{\jobname.bib} \nocite{*}  \newbibmacro{name:newformat}{\printnames{authors}} \DeclareNameFormat{newformat}{%   \usebibmacro{name:newformat}{\textbf{#1}}{\textbf{#3}}{#5}{#7}%   \usebibmacro{name:andothers}% }  \begin{document}    \section*{Testing the format here}    I am citing   \citename{Doe12}[newformat]{author}    \printbibliography % to make sure that the .bib file is properly formated.  \end{document}   Here, I just used \textbf for the sake of the example, but I ultimately want to use other functions, say upper case or palindrome!  However, I get this weird error message:  ! Illegal parameter number in definition of \blx@defformat@d. &lt;to be read again&gt;                     3 l.20   \usebibmacro{name:last}{#1}{#3                                      }{#5}{#7}%   which seems to indicate that the second option {#3} is not expected. I am a bit at a lost here; I checked the biblatex package and \usebibmacro{name:last} must be given 4 parameters, isn't it? Please advise. Is there a case insensitive jQuery :contains selector? Is there a case insensitive version of the  jQuery selector or should I do the work manually by looping over all elements and comparing their .text() to my string? What does ':' (colon) do in JavaScript? I'm learning JavaScript and while browsing through the jQuery library I see : (colon) being used a lot. What is this used for in JavaScript?   // Return an array of filtered elements (r) // and the modified expression string (t)    return { r: r, t: t }; What is the difference between 'git pull' and 'git fetch'? What are the differences between git pull and git fetch? Why there is a $\frac{1}{2}$ in the distance formula $d=\frac{1}{2}at^2$? I'm preparing for my exam, but I have difficulties in perceiving why there is a $\frac{1}{2}$ in the distance formula $d=\frac{1}{2}at^2$? What's the English equivalent of the Japanese saying, “A fart ruins 100 days of sermons by the priest (bishop)”? I was amused by the expression "Paid a penny and only farted" (related by @FumbleFingers), which suggested a similar Japanese saying: 大山鳴動鼠一匹 - "Find only a small mouse coming out after hearing rumblings and experiencing shaking of the big mountain."  It’s humorous. I chuckled when I read it, and I wrote to the author that I love it.  With that said, the word "fart" reminds me of another Japanese proverb, 百日の説法屁一つ- "A fart ruins 100 days of sermons by the priest", which means that a small mistake (the sound of a fart in front of the congregation) brings all his efforts to naught.  It can’t happen in big Western churches. But it could well happen in a small wooden temple where a priest would preach to 30 – 50 provincial followers. Can you imagine how the proud priest being called 'the venerable' would be embarrased when his loud fart is heard by all his followers during his serious sermon and they start to giggle here and there in the hall?  We also have a similar saying" 九仞の功を一簣に欠く, which literally means "Collapse a 9- meter sand tower with the last pile of sand."  What is the English equivalent of the saying "A fart ruins 100 days of sermons by the priest (bishop): a small mistake depreciates the value of (vitiates) all labor and effort to naught," preferably in such a humorous way? Is there a way to find out if I need a transit visa for a layover in the UK? How do I install the latest Python 2.7.X or 3.X on Ubuntu? I want to install the latest Python tarball on Ubuntu, downloaded from .  Is this is a correct way to install?  ./configure make make install   If not, how do I do that? What are some good books for learning general relativity?