Prove $\cos(x)<\frac{\sin^2(x)}{x^2}$ I have to prove that $\cos(x)&lt;\frac{\sin^2(x)}{x^2}$ for $0&lt;x&lt;\frac\pi2$. By using the MVT I got that $\frac{\sin^2(x)}{x^2}=\cos^2(c)&gt;\cos^2(x)$ for a $c\in[0, x]$ but this doesn't help since $\cos^2(x)&lt;\cos(x)$ because $0&lt;\cos x&lt;1$. The derivative didnt help either, since the derivatives of both $\cos(x)x^2-\sin^2(x)$ and $\cos(x)-\frac{\sin^2(x)}{x^2}$ don't have roots that I can easily find/I don't see a way to prove that they don't have roots. Any help is appreciated!
Prove $\sin(x)\tan(x) > x^2$ for $x \in ( \,0, \frac{\pi}{2}) \,$ Prove $\sin(x)\tan(x) &gt; x^2$ for $x \in ( \,0, \frac{\pi}{2}) \,$  So I did the following: Let $f(x) = \sin(x)\tan(x) - x^2$.  Then of course $f(0) = 0$ and I want to show that for $x \in ( \,0, \frac{\pi}{2}) \,$ $f'(x) &gt; 0$ but I don't really know where to go from here as I can't get anything reasonable done with the derivative.  I was thinking of approximating it using the following inequality $\sin(x) &lt; x &lt; \tan(x)$.   Is that a good direction?
Evaluate $\int_0^1 \frac{\log \left( 1+x^{2+\sqrt{3}}\right)}{1+x}\mathrm dx$ I am trying to find a closed form for  $$\int_0^1 \frac{\log \left( 1+x^{2+\sqrt{3}}\right)}{1+x}\mathrm dx = 0.094561677526995723016 \cdots$$  It seems that the answer is $$\frac{\pi^2}{12}\left( 1-\sqrt{3}\right)+\log(2) \log \left(1+\sqrt{3} \right)$$  Mathematica is unable to give a closed form for the indefinite integral.  How can we prove this result? Please help me.  EDIT  Apart from this result, the following equalities are also known to exist: $$\begin{align*} \int_0^1 \frac{\log \left( 1+x^{4+\sqrt{15}}\right)}{1+x}\mathrm dx &amp;=\frac{\pi^2}{12} \left( 2-\sqrt{15}\right)+\log \left( \frac{1+\sqrt{5}}{2}\right)\log \left(2+\sqrt{3} \right) \\ &amp;\quad +\log(2)\log\left( \sqrt{3}+\sqrt{5}\right) \\ \int_0^1 \frac{\log \left( 1+x^{6+\sqrt{35}}\right)}{1+x}\mathrm dx &amp;= \frac{\pi^2}{12} \left( 3-\sqrt{35}\right)+\log \left(\frac{1+\sqrt{5}}{2} \right)\log \left(8+3\sqrt{7} \right) \\ &amp;\quad +\log(2) \log \left( \sqrt{5}+\sqrt{7}\right) \end{align*}$$  Please take a look . Textbooks on linear regression with least squares I have a follow up question to this question:   My question is: when should you add a comma when starting a letter/email with     Hello John,      It was fun to see ...   I am pretty sure that a comma goes after the name, but lately I have seen several people (academics) write      Hello, John,       It was fun to see ...   with a comma after the Hello.  I am wondering when a comma there is appropriate.    I see this question  but I am wondering about the situation where you have a comma after the name also. New question asking form sometimes leads to accidental double posts of questions I've noticed that with the new question asking form, sometimes, attempting to submit a question will post two identical copies of the same question.  I've  (), but at the time, I assumed it was just a temporary fluke. As far as I remember, I just reviewed the question, saw no errors, and submitted it. (That's all the info I have.)  However, I just saw this happen to someone else, so it can't just be a fluke on my end. , and . The author claims there that they did indeed double click the "Review" button, but their question didn't go through since it didn't have a required tag, but when they tried to submit the question with the required tag, they only clicked it once.  Also, now that I think about it, it could not have been just a temporary client-side issue on both our ends. The system is supposed to  if one tries to post more than one question in 30 seconds. Neither of us got a CAPTCHA. This means that the server accepted the double post, and so it's a server issue.  This has the effect of denying users badges that they would have otherwise qualified for, since a single negatively-received, closed, or deleted question . This means that even if the double post is deleted and the original is upvoted, the user won't qualify for that day since they've made a "bad" question on that day. (While they may be able to later reverse this 60 days later by undeleting and re-deleting their question - questions deleted more than 60 days later aren't considered "deleted" for the check - they often won't be able to, since most of the time, users will have downvoted the double post, which also disqualifies one from the badges.)  Can an employee go through server logs to see what happened in both these cases, and add some safeguards to prevent accidental double submissions of questions? This seems to have come up when the new asking form was deployed.  Update: There was another case just now:  was . As both have the exact same time of submission, this is an instance of this bug, since as I said above, this thing is ordinarily prevented by a CAPTCHA. Insert missing dates within table I have a table with a column for ID, Date and Frequency of occurrence for that particular date and ID.  ID    date          Frequency 1     2014-05-18      5 1     2014-05-19      4 1     2014-05-20      25 2     2014-05-20      7 3     2014-05-18      4 3     2014-05-20      1 4     2014-05-18      6   I'm looking to insert the dates for each ID that aren't currently there (assuming that the date ranges from 2014-05-18 to 2014-05-20) and add a value of 0 in Frequency for those.  For example, you can see that  ID 1 has an observation for all three dates;  ID 2 is missing 2014-05-18 and 2014-05-19;   ID 3 is missing 2014-05-19;   ID 4 is missing 2014-05-19 and 2014-05-20.  Therefore, the output should look like this  ID    date          Frequency 1     2014-05-18      5 1     2014-05-19      4 1     2014-05-20      25 2     2014-05-18      0 2     2014-05-19      0 2     2014-05-20      7 3     2014-05-18      4 3     2014-05-19      0 3     2014-05-20      1 4     2014-05-18      6 4     2014-05-19      0 4     2014-05-20      0   I've attempted the following from . However, that will only add dates with 0 Frequency that aren't present throughout the entire table instead of on an ID level.   DECLARE @s DATE = '20140518', @e DATE = '20140520';  SELECT c.date, Frequency = COALESCE(a.Frequency,0) FROM dbo.Calendar AS c LEFT OUTER JOIN dbo.table AS a ON c.date = s.[date] WHERE c.d &gt;= @s AND c.d &lt; DATEADD(DAY, 1, @e); Is there a string math evaluator in .NET? If I have a string with a valid math expression such as:  String s = "1 + 2 * 7";   Is there a built in library/function in .NET that will parse and evaluate that expression for me and return the result? In this case 15. Is MacBook Pro warming during gaming normal? I have a 2010 MacBook Pro MC371 (Core i5, GeForce 320M, 4Gb RAM).   When I launch a game either new one (i.e. Starcraft 2) or rather old (Unreal Tournament 2004, bootcamped) my MacBook gets rather warm, even hot in the top left corner, around the place where magsafe charger is plugged in. The temperature is more than 40C. The fact that confuses me is that this warming occures even when I play old games.  Is that OK? How to prove that the center of the dihedral group $D_{2n}$ is $\{1,r^{n}\}$ and the center of $D_{2n-1}$ is $\{1\}$?  I don't know how to prove it in this general case. Rules for refreezing food What are the general rules regarding when can you refreeze food that was frozen and then thawed in the fridge? and what are the reasons behind them? (Hygiene? Food quality?). Let's assume for the sake of the argument that the items were in a non-frozen state only a few hours.  To be more specific, I'm interested in the application of these rules for  - Ginger garlic paste - Dough - Indian Curries - Green Chilly paste - Frozen green peas Cardinality of a totally ordered union How to give different permissions to multiple groups of users? I installed Ubuntu 12.04 server and want to give full folder access say example: "Folder01" to group "Group01' and Limit "Group02" for read access only. How I can do this. Why do I get "Permission denied (publickey)" when trying to SSH from local Ubuntu to a Amazon EC2 server? I have an instance of an application running in the cloud on Amazon EC2 instance, and I need to connect it from my local Ubuntu. It works fine on one of local ubuntu and also laptop. I got message "Permission denied (publickey)" when trying to access SSH to EC2 on another local Ubuntu. It's so strange to me.  I'm thinking some sort of problems with security settings on the Amazon EC2 which has limited IPs access to one instance or certificate may need to regenerate.  Does anyone know a solution? Title does not appear when LaTeX code is compiled Two monitors when only having one in 13.10 I just did a fresh install of Ubuntu 13.10 64bit, and it says that I have 2 monitors, when I actually only have one    I have a Dell XPS 15 with Nvidia graphics card if that helps in anyway PHP script not executing on Apache server If a normal subgroup $N$ of order $p$($p$ prime) is contained in a group $G$ of order $p^n$,then $N$ is in the center of $G$. If a normal subgroup $N$ of order $p$($p$ prime) is contained in a group $G$ of order $p^n$,then $N$ is in the center of $G$.   I want to use induction to prove this:  It is trivial when $G=p(n=1)$, assume $G=p^n$, $N$ is a normal subgroup in $G$ with order $p$. Since $G$ is p-subgroup, $G$ has a normal subgroup $H$ with order $p^{n-1}$,according to assumption, $N \subset C(H)$. But I don't know how to go on, is $C(H)=C(G)$?Can induction work in this case? How can I remove the qed symbols from all lemma proofs? the following code will only make the qed not appear in the second proof. However, it it possible to make the qed symbol never appear at the end of the proofs without having to specify it locally every time?  \documentclass{article} \usepackage{amsthm} \begin{document} \begin{proof} This has the QED symbol. \end{proof} \begin{proof}\renewcommand{\qedsymbol}{} This hasn't. \end{proof} \begin{proof} And this has it again. \end{proof} \end{document} Nested List Indices Open a direct file on the hard drive from Firefox (file:///) I have a small PHP system I created for myself. This system contains a few .doc and .docx file documents. I want to be able to open them directly from the browser without downloading them.  I created a link: &lt;a href="file:///c:/mysite/myfile.docx"&gt;myfile&lt;/a&gt;. But Firefox does not open the link. href="http://localhost/myfile.docx" works fine - it offers me to download the file.  Any ideas on how I can do it? Do I need to change something in the Firefox options or security settings? So this is a question with which I have been stuck for a while and it all revolves around the concept of volt.  I have posted a LOT of questions on physics forums while trying to understand the concept of volt and potential difference. This is hopefully going to be the last one.  What do you exactly mean when you say that a battery is 2.5 V. I know that it says that the potential difference between the 2 terminals of the battery is 2.5 V  but then what is the potential of the 2 ends of the battery? Is there a unit of potential and not just potential difference?  Also volt = work/charge but how can work/charge express the difference of potentials across the 2 terminals?