So, I just tried using Pythagoras theorem to prove: $\frac{d}{dx}\sin x= \cos x$ but it got me nowhere. Here is my work. $$\sin x= p/h.$$; where p and h are lengths of perpendicular and hypotenuse in a right angled triangle. Differentiating w.r.t $x$, we get $$\frac{d}{dx}\sin x= \frac{1}{h^2} \left( h\frac{dp}{dx}-p\frac{dh}{dx}\right)\tag{i}$$ We know $h^2=p^2+b^2$. (Where b is the length of the base of the right angled triangle.) Differentiating w.r.t $x$, $$h\frac{dh}{dx}=p\frac{dp}{dx}$$ (If we put $b$ constant). Substituting in $(i)$, \begin{align} \frac{d}{dx} \sin x&amp;= \frac{1}{h^2} \left(h\frac{dp}{dx}-\frac{p^2}{h}\frac{ dp}{dx}\right)\\        &amp; = \frac{1}{ph^2} \left(b^2\frac{dp}{dx}\right) \tag{ii} \end{align} Now, I lost track here because I didn't know how is that equal to $\cos x$. Can anyone help me continue from here? (Note: I know how to derive this from first principle. I just wanted to try using Pythagoras theorem and trigonometric identities. I kept the base length constant because it appeared to me that if I changed the x, base length won't be effected. But as I can see in the answer, the hypotenuse remains constant. However, can we keep the base constant and end up getting the same result? Maybe not visualising a circle of radius h but just a right angled triangle whose base remains constant w.r.t the reference angle?  Ps: The called duplicate one is out of my understanding. I request a simpler version.
One of the first things ever taught in a differential calculus class:   The derivative of $\sin x$ is $\cos x$. The derivative of $\cos x$ is $-\sin x$.   This leads to a rather neat (and convenient?) chain of derivatives:   sin(x) cos(x) -sin(x) -cos(x) sin(x) ...   An analysis of the shape of their graphs confirms some points; for example, when $\sin x$ is at a maximum, $\cos x$ is zero and moving downwards; when $\cos x$  is at a maximum, $\sin x$ is zero and moving upwards.  But these "matching points" only work for multiples of $\pi/4$.  Let us move back towards the original definition(s) of sine and cosine:  At the most basic level, $\sin x$ is defined as -- for a right triangle with internal angle $x$ -- the length of the side opposite of the angle divided by the hypotenuse of the triangle.  To generalize this to the domain of all real numbers, $\sin x$ was then defined as the Y-coordinate of a point on the unit circle that is an angle $x$ from the positive X-axis.  The definition of $\cos x$ was then made the same way, but with adj/hyp and the X-coordinate, as we all know.  Is there anything about this basic definition that allows someone to look at these definitions, alone, and think, "Hey, the derivative of the sine function with respect to angle is the cosine function!"  That is, from the unit circle definition alone.  Or, even more amazingly, the right triangle definition alone.  Ignoring graphical analysis of their plot.  In essence, I am asking, essentially, "Intuitively why is the derivative of the sine the cosine?"
Table of Contents for set of PDF files How to make Table of Contents for a set of PDF files?  Here's my code!  \documentclass{book} \usepackage{pdfpages} \usepackage[pdfauthor={&lt;somename&gt;},pdftitle={&lt;sometitle&gt;},pdftex]{hyperref} \usepackage{fancyhdr} \usepackage{tocloft} \usepackage[margin=0.5in,bottom=0.75in,top=0.75in]{geometry}   \pagestyle{fancy} \renewcommand{\headrulewidth}{0.4pt} \fancyhead[L]{X Y Z} \fancyhead[R]{} \begin{document}  \tableofcontents \newpage  \clearpage \phantomsection \includepdfset{pages=-,pagecommand=\thispagestyle{fancy}} \addcontentsline{toc}{chapter}{SAMPLE PDF 1\\ \textit{description}} \includepdf[pages=-]{template1.pdf}  \clearpage \phantomsection \addcontentsline{toc}{chapter}{SAMPLE PDF 2\\ \textit{description}} \includepdf[pages=-]{template2.pdf}  \clearpage \phantomsection \addcontentsline{toc}{chapter}{SAMPLE PDF 3\\ \textit{description}} \includepdf[pages=-]{template3.pdf}  \clearpage \phantomsection \addcontentsline{toc}{chapter}{SAMPLE PDF 4\\ \textit{description}} \includepdf[pages=-]{template4.pdf}  \clearpage \phantomsection \addcontentsline{toc}{chapter}{SAMPLE PDF 5\\ \textit{description}} \includepdf[pages=-]{template5.pdf}  \end{document}   Here's the output.    My problem is that the page number is not aligned with the title, it is aligned with the description. Also I'd like to put dots.   My desired output looks like this: If you make a search for something that returns more than 10,000 items in result. For example, search for  return 161,338 results on MSE. But if you select a page where 10,000+ result should be shown:        In the edge case  shows zero results:        But this is actual for any page where per_page * page_number is more than 10,000.  Though this does not happen for Stack Overflow in English. It looks like all other sites are affected.  Related bug report on ruSO.meta: I have an input type="image". This acts like the cell notes in Microsoft Excel. If someone enters a number into the text box that this input-image is paired with, I setup an event handler for the input-image. Then when the user clicks the image, they get a little popup to add some notes to the data.  My problem is that when a user enters a zero into the text box, I need to disable the input-image's event handler. I have tried the following, but to no avail.  $('#myimage').click(function { return false; }); Upgrading from a modern smartphone camera without going all in on a DSLR kit What is the difference between Halt and Shutdown commands? I would like to transfer files between two remote hosts using on local shell, but it seems rsync doesn't support synchronisation if two remotes are specified as follow:  $ rsync -vuar host1:/var/www host2:/var/www The source and destination cannot both be remote.   What other workarounds/commands I could use to achieve similar results? I am currently using 19.04 and I got "No new release found for 19.10" How can I enable Hardmode? What items do I need to make my world Hard Mode? I have a bash script which must call another bash script which varies per-user in order to get some configuration information, like so:  somescript:  #!/bin/bash ${HOME}/.ssconfig echo "Hello, ${VARIABLE}!"   ~/.ssconfig  #!/bin/bash VARIABLE=world   Which, instead of giving me Hello, world! as expected, gives me Hello, ! How can I pass variables up from a script to the one that called it? I know the usage of     This is a new car.   This is singular.     These are some books for you.   These is plural.  Shall we use   There exists some people who agree with me. There exist some people who agree with me.   Which one is the correct usage and why? A probability theory question about independent coin tosses by two players Say Bob tosses his $n+1$ fair coins and Alice tosses her $n$ fair coins. Lets assume independent coin tosses. Now after all the $2n+1$ coin tosses one wants to know the probability that Bob has gotten more heads than Alice.    The way I thought of it is this : if Bob gets $0$ heads then there is no way he can get more heads than Alice. Otherwise the number of heads Bob can get which allows him to win is anything in the set $\{1,2,\dots,n+1\}$. And if Bob gets $x$ heads then the number of heads that Alice can get is anything in the set $\{0,1,2,..,x-1\}$. So\begin{align}P(\text{Bob gets more heads than Alice})&amp;= \sum_{x=1}^{n+1} \sum_{y=0}^{x-1} P( \text{Bob gets x heads }\cap \text{Alice gets y heads }) \\[0.2cm]&amp;= \sum_{x=1}^{n+1} \sum_{y=0}^{x-1} \left(C^{n+1}_x \frac{1}{2}^{x} \frac{1}{2}^{n+1-x}\right)\left( C^n_y \frac{1}{2}^y \frac {1}{2}^{n-y}\right)\\[0.2cm]&amp; = \sum_{x=1}^{n+1} \sum_{y=0}^{x-1} \frac{C^{n+1}_x C^n_y}{2^{2n+1}}\end{align}   How does one simplify this?      Apparently the answer is $\frac{1}{2}$ by an argument which looks like this, Since Bob tosses one more coin that Alice, it is impossible that they toss both the same number of heads and the same number of tails. So Bob tosses either more heads than Alice or more tails than Alice (but not both). Since the coins are fair, these events are equally likely by symmetry, so both events have probability 1/2. Manga where the protagonist need to solve girl's quests Difference between static class and singleton pattern? What real (i.e. practical) difference exists between a static class and a singleton pattern?  Both can be invoked without instantiation, both provide only one "Instance" and neither of them is thread-safe. Is there any other difference? Can I have multiple primary keys in a single table? Can I have multiple primary keys in a single table? Java multiline string Beamer: PDF presentation viewer to play embedded videos and show elapsed time I have been suffering from finding a PDF presentation viewer for beamer output on Windows that can play embedded videos (by media9 package) and show the elapsed presentation time. I have tried the following: Adobe Reader with tdclock package:  Pros: plays videos and loads tdclock properly Cons: sluggish and crashes frequently  Foxit Reader with tdclock package:  Pros: smooth viewer and plays videos  Cons: can't load tdclock clock/stopwatch despite enabling javascript   Impressive:  Pros: fast and has internal clock  Cons: can't play embedded videos   Dual-Screen PDF Viewer:  Pros: very fast, has internal clock and dual screen support  Cons: can't play embedded videos   Is there any other solution that can properly play embedded videos and show elapsed presentation time flawlessly and smoothly on Windows?  P.S. when I mean tdclock javascript is not loaded, those empty red boxes show up instead of the clock/stopwatch,  and when embedded video can't be loaded, a white page shows up. Greek Text (babel) and English Code Text (listings): Problem I use the babel package in order to write in greek with font "kerkis".  I want to add code in english ("courier new" font) to my latex document but I get some difficulties.  Can anybody help me, please. Thank you in advance.  \documentclass[12pt]{article} \usepackage[english,greek]{babel} \usepackage[utf8x]{inputenc} \usepackage{graphicx} \usepackage{fancybox} \usepackage{verbatim} \usepackage{courier} \usepackage{kerkis} \usepackage{float} \usepackage{geometry} \usepackage{fancyhdr} \usepackage{titlesec} \usepackage{tocloft} \usepackage{hyperref} \usepackage{afterpage} \usepackage{color}    \usepackage{listings}  \renewcommand{\lstlistingname}{Κώδικας}  \lstset{  %matlab   backgroundcolor=\color{white},     basicstyle=\normalsize,    % COURIER NEW?        commentstyle=\color{green},    % comment style   escapeinside={\%*}{*)},           extendedchars=true,               frame=single,                              keywordstyle=\color{blue},       % keyword style   numbers=left,                     numbersep=5pt,                     numberstyle=\normalsize\color{black},    rulecolor=\color{black},           stepnumber=1,                      stringstyle=\color{magenta},     % string style }   \begin{document}  \selectlanguage{greek} Ελληνικά: \textlatin{this word means greek}  \selectlanguage{english} Now I write in english  \selectlanguage{greek} Ελληνικά  %\selectlanguage{english}???  \begin{lstlisting}[caption = \textlatin{example1.m}] { % Entropy E = 0; for i = 1:J     if p(i) ~= 0         E = E + p(i)*log2(1/p(i));     end end } \end{lstlisting}  \end{document} Grep: unexpected results when searching for words in heading from man page What do you call a person that won’t act? What would you call a person that just stands by and won't act? Is kind of acting like a spectator when in reality that person should do something?  I know there is a word for this because I knew what it was a few hours ago but now I can’t remember it. I've got a NI Traktor Audio 2 sound card I'd like to get working with Mixxx, but I don't seem to have any success so far. The card seems to be supported in Ubuntu out of the box, and it seems to work well with ALSA, but I cannot select it in Mixxx and I'm not sure whether is something I can do to make it see it (e.g. through).  So for testing purposes, on the command line I can execute aplay to check each channel:  aplay -D plughw:2,0,0 test.wav   and  aplay -D plughw:2,0,1 test.wav   That works well: I can hear the test sound on each channel (A and B)  The relevant output for this card using aplay -D is:  hw:CARD=TraktorAudio2,DEV=0      Traktor Audio 2, Traktor Audio 2      Direct hardware device without any conversions plughw:CARD=TraktorAudio2,DEV=0      Traktor Audio 2, Traktor Audio 2      Hardware device with all software conversions   From those 2 items on the output, plughw is the one that works, whereas hw doesn't.  The problem is that Mixxx only lets me select the hw card, which does not seem to work. I'd like to be able to select the plughw card, which is the one that's working.  I've tried suspending Pulse Audio as suggested by , but that did not seem to work. I also played around with an .asoundrc file similar to the one  (which was for an older model of this sound card), but also without success.  Any ideas on how to make Mixxx show the right card (and channel, while we are at it)?  Thanks!