$\frac{ \sqrt{1+\sin(x)}+\sqrt{1+\cos(x)} }{ \sqrt{1-\sin(x)}+\sqrt{1-\cos(x)} }=\sqrt2+1\,\forall x\in [0,\pi/2]$ the fraction of square roots may be simplified using double angle formulas.. though i wouldn't recommend showing the derivative to be 0...
Here is the question:     $$\frac{\sqrt{10+\sqrt{1}}+\sqrt{10+\sqrt{2}}+\cdots+\sqrt{10+\sqrt{99}} }{\sqrt{10-\sqrt{1}}+\sqrt{10-\sqrt{2}}+\cdots+\sqrt{10-\sqrt{99}}} = \;?$$   ()  I think we need to simplify it writing it in summation sign as you can see here:  $$\frac{\sum\limits_{n=1}^{99} \sqrt{10 + \sqrt{n}}}{\sum\limits_{n=1}^{99} \sqrt{10 - \sqrt{n}}}$$  or in Wolfram Alpha input in comments.  I can compute it too! It's easy to write a script for this kind of question. I need a way to solve it. How would you solve it on a piece of paper?
I find that at the end of a long day, I sometimes have a large number of programs running. All which I will need to use tomorrow. Normally, this isn't an issue, I can simply lock the machine and come back tomorrow. My problem arrises when windows update launches in the middle of the night and force restarts my computer. That in turns closes all my open software. I of course save everything regularly so I don't loose anything, but I waste time reopening all of those resources whenever there is a restart.   [EDIT]  I should clarify that I still want to be able to restart my computer when an update comes down. Preventing the restart only delays the problem until later. I should have been more specific in that I want to be able to recover my working environment after a restart for any reason. Things like scheduled maintence, power loss, updates, and software installs.  [EDIT]  I can't simply have them setup to launch at startup becasuse those files change from week to week. So I need something that monitors what I have open, and gives me the option to "recover" those software sessions when I log back in.  Anyone have any suggestions on what I can do? I'd even be willing to purchase software to do this for me if that is the only option.  Thanks If a character is drunk, what are the penalties associated to it? Are there any benefits to it? I do remember a special prestige class for monks about it, but I don't remember anything specific in the core rulebook. Every finite group is isomorphic to some Galois group for some finite normal extension of some field. (Built-in) way in JavaScript to check if a string is a valid number Cycle Normal Pass in Camera Space I would change the font of section and subsection of my document. I have created it using book class. I would use this font:    According to Adobe pdf propriety, this font is AvantGarde-Demi. How can I use it in Latex. what command I should use to recreate this in latex. I think I should use titlesec package right? Lost my APFS partition after using EaseUS partition manager on Bootcamp I used macOS mojave(APFS partition) and Windows10(Bootcamp). And I tried to expand Bootcamp partition with moving Windows OEM partition by EaseUS partition master in windows. After this, I can’t find APFS partition installed macOS.  I use MacBook Pro 2016.  I think “disk0s2” was APFS partition, but now it is “Windows Recovery” partition.  Do you have any ideas to solve this? (I am Japanese, so I'm sorry for my poor English)  I tried “gpt show” and “diskutil list” on macOS Recovery   Additional information  I tried dd if=/dev/disk0s2 count=1 bs=512 command in a Terminal app of macOS Utility from network boot, and the terminal says,  -bash-3.2# dd if=/dev/disk0s2 count=1 bs=512 | vis -c 1+0 records in 1+0 records out 512 bytes transferred in 0.005472 secs (93568 bytes/sec) )us??-ҵ?\^A\0\0\0\0\0\0\0?\M^F\^Z\0\0\0\0\0\^A\0\0\M^@\0\0\0\0NXSB\0\^P\0\0?\M^B?\^B\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\^B\0\0\0\0\0\0\0"\^R???\M^_@}\M^Bvy57 ?(h?8\0\0\0\0\0?\M^F\^Z\0\0\0\0\0\^X\^A\0\0 l\0\0?\M^J\0\0\0\0\0\0\^R\r\0\0\0\0\0\0\a\0\0\0004*\0\0\^E\0\0\0\^B\0\0\0\^S*\0\0!\0\0\0\M^Qr8\0\0\0\0\0\M^U?\b\0\0\0\0\0\^A\^D\0\0\0\0\0\0\0\0\0\0d\0\0\0\^C\^D\0\0\0\0\0\0\M^T\M^N\^A\0\0\0\0\0\M^V\M^N\^A\0\0\0\0\0?0\^B\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0-bash-3.2#   . Man creates miniature population on different time-scale; uses them to invent the 'absolute protection' dome 'sudo su -' vs 'sudo -i' vs 'sudo /bin/bash' - when does it matter which is used, or does it matter at all? Theme Activate Hook What techniques can be used to define a class in JavaScript, and what are their trade-offs? I prefer to use OOP in large scale projects like the one I'm working on right now. I need to create several classes in JavaScript but, if I'm not mistaken, there are at least a couple of ways to go about doing that. What would be the syntax and why would it be done in that way?  I would like to avoid using third-party libraries - at least at first. Looking for other answers, I found the article  that discusses object-oriented programming in JavaScript. Is there a better way to do inheritance? The easiest thing to do is [to] call for a taxi How can I modify the size of swap with LVM partitions? Setting CSS pseudo-class rules from JavaScript I'm looking for a way to change the CSS rules for pseudo-class selectors (such as :link, :hover, etc.) from JavaScript.  So an analogue of the CSS code: a:hover { color: red } in JS.  I couldn't find the answer anywhere else; if anyone knows that this is something browsers do not support, that would be a helpful result as well. Should I change the URLs in the XML sitemap after 301 redirecting them? I am using 301 redirects for all URLs.  Should I change sitemap URLs now?  The sitemap currently contains the following URLs:   example.com/cat/1.html example.com/cat/2.html example.com/cat/3.html   Should I change the sitemap's URLs to a new destination?   example.com/cat/year/1.html example.com/cat/year/2.html example.com/cat/year/3.html Do badges get calculated in real time, or do they get calculated by an external service? How long does it take for a badge to be awarded, once its requirements are met?  Does Stack Overflow use any external services to do things, like maintenance, e-mailing or badge calculation? Need help with SQL Server PIVOT I know about the PIVOT feature in SQL Server and I've read a few other questions and answers on here but I just can't seem to grasp it in order to do what I need. I'm trying to run some calculations and I have the following:  TABLE office_settings column office_id (bigint not null) column setting_value (NVARCHAR(500) NOT NULL)  office_id    setting_value -------------------------------------------- 4             0.2 4             0.2 4             2 4             3 5             0.2 5             0.2 5             2 5             3   I'd like to convert this using PIVOT (assuming that's the right tool for the job) to this, basically "grouping by" the office_id column, so:  office_id    value_1    value_2    value_3   value_4 ------------------------------------------------------     4         0.2         0.2         2         3     5         0.2         0.2         2         3   I should also mention that I need to CAST the setting_value column values to DECIMAL(4,2) in the process of this - in case that has any bearing on the T-SQL required to accomplish it.  Any ideas? Can I choose to delay my turn in combat? Creating Indexes on DB with Hibernate @Index Annotation How can I define a custom listing environment? I am trying to define an enviroment for a specific programming language, F#. Here's my code:  \definecolor{bluekeywords}{rgb}{0.13,0.13,1} \definecolor{greencomments}{rgb}{0,0.5,0} \definecolor{turqusnumbers}{rgb}{0.17,0.57,0.69} \definecolor{redstrings}{rgb}{0.5,0,0}  \lstdefinelanguage{FSharp}     {morekeywords={let, new, match, with, rec, open, module, namespace, type, of, member, and, for, in, do, begin, end, fun, function, try, mutable, if, then, else},     keywordstyle=\color{bluekeywords},     sensitive=false,     morecomment=[l][\color{greencomments}]{///},     morecomment=[l][\color{greencomments}]{//},     morecomment=[s][\color{greencomments}]{{(*}{*)}},     morestring=[b]",     stringstyle=\color{redstrings}     }  \newenvironment{fslisting}   {     \lstset{         language=FSharp,         basicstyle=\ttfamily,         breaklines=true,         columns=fullflexible}     \begin{lstlisting}   }   {     \end{lstlisting}   }   Then I'm trying to use it as such:  \begin{fslisting} type 'a Process = 'a Signal IObservable \end{fslisting}   But it doesn't work. The output log says that the listing is not terminated - or is terminated by the document. So the rest of the document is also formatted as a listing, which is of course not the intention.  What am I doing wrong here?