Is there any other way to prove it rather than proving it using the formula by definition ?
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?"
Change email login to another id My original email id got compromised and now I want my new email id to be the main id for all SE sites . How to delete original id from the system as someone else might login from it ? How can I show or hide boot messages when Ubuntu starts? Is there a way to easily turn on/off showing the boot messages (loading the services) when Ubuntu starts? Is it something in Grub2?  I am running 10.04. US and Canada border crossing and computer privacy: What do I need to know? I've read stories about border officials searching laptops when crossing the border in one direction or the other. Does this really happen, typically speaking?  As a US citizen, can I be forced to give them my laptop's password for any/no reason? I don't believe they are allowed to deny entry to the US to a US citizen? ....but who knows these days?  Canada can deny me entry for any reason, so that's fine. But in practice, does this even happen in either crossing direction?  EDIT - I've now made the trip. They didn't so much as glance in the direction of my laptop. My biggest issue turned out to be lack of international wireless data service! Google Chrome Address Bar Font has gone too large, how can we reduce it Running Unity on my desktop (2*20" LCD displays), I initiated a standard daily update on my Ubuntu 14.04 LTS today and Google Chrome updated itself to Version 35.0.1916.114.  The URL address bar input now shows IMO a too larger font size which then proceeds to auto show in green. I prefer the smaller font size and have no need for 'gooey slime green' auto complete font colours. Any advice on how to revert back to smaller, generic Address font styling.  I use both FF and Chrome for numerous tasks, Chrome being my primary browser and I've checked my system Unity 'default' theme (Ambiance! LOL.. yes I'm tired of that also).  Thanks for sharing. How to validate an email address in JavaScript For all types of derivative works, namely:   Movies Novels Comic books Other written works (including RPG materials) Cartoons Video games Derivative movies (such as the infamous Ewok movies and even more infamous )   According to what rules does something become "canon" and who is responsible for the evaluation of a published work? Prove that "injective function $f:X\to Y$ exists" and "surjective function $g:Y\to X$ exists" is logically equivalent. Prove that   injective function $f:X\to Y$ exists surjective function $g:Y\to X$ exists   is equivalent.$(x,y\neq \emptyset)$   My approach  First,prove left to right.  If injective function $f:X\to Y$ exists, a set A can be defined like this. $$A=\{f(x):x\in X\}$$ then it is sure that $A\subset Y$, and $|A|=|X|$  And, if we define function $g$ like $$g(y)=\begin{cases}x &amp; \text{if $y\in A$ and } f(x)=y,\\x_0&amp; \text{if }y\notin A.\end{cases}$$ ($x_0$ is a fixed element of $X$)  Then, $g$ is a surjective function, so surjective function $g:Y\to X$ exists.  Now,prove right to left.  If surjective function $g:Y\to X$ exists, $\forall x \in X$, we can select a element $y\in Y$ that $g(y)=x$, which is clear that we can pick a different $y\in Y$ for $\forall x \in X$  if we define function $f$ as $f(x)=\text{one element in Y that g(that element)=$x$}$, f is a injective function, so injective function $f:X\to Y$ exists.    But, I am not sure that my approach is right, especially on proving right to left. Please check whether my approach is right, ways to improve it,  and some other ways of proving this question. From :     Importantly, the DDH assumption does not hold in the multiplicative group $\mathbb {Z} _{p}^{*}$, where $p$ is prime. This is because given $g^{a}$ and $g^{b}$, one can efficiently compute the Legendre symbol of $g^{ab}$, giving a successful method to distinguish $g^{ab}$ from a random group element.   The only way I can think of to distinguish $g^c$ from $g^{ab}$ is:    If $\left(\frac{g}{p}\right)=-1$, then $\left(\frac{g^{ab}}{p}\right) = 1$ if and only if $a$ or $b$ is even (which is $3/4$ of the times). So if $\left(\frac{g^c}{p}\right) =-1$, probably $g^c\neq g^{ab}$. Nothing is certain.   How'd do you compute $\left(\frac{g^{ab}}{p}\right)$ given $g^a$ and $g^b$? Standard error of a ratio I have a linear regression and two estimates, say A and B and their standard errors. I need to find the standard error of a ratio A/B [or A/(1-B)].  I guess the main problem is that I don't know the correlation between A and B. I also guess that makes this unsolvable, since the correlation plays a major role. Is this correct?  If I knew it, what would be the way to calculate the standard error? Including +/- signs to battery in Circuitikz I'm attempting to draw a battery (using the battery1 component) that includes positive and negative signs at the appropriate ends. A  that was answered here (using version 2.4) shows exactly what I mean:    Only, of course, I want to be using a battery1 symbol instead. So, I have my code as follows:  \usepackage[americanvoltages,fulldiodes,siunitx]{circuitikz} \begin{document} \begin{circuitikz}     \draw (0,0) to [battery1=\SI{9}{V}] (3,0)                 to [pC,v=$v_C$] (6,0); \end{circuitikz} \end{document}   And I end up with this:    Notice the lack of +/- signs (I know how circuits work so I don't need to be actually shown these signs, but for the sake of copying my class notes diagrams I want these signs to be included in my latex diagram).  In either case, I think it may have something to do with the version of Circuitikz I'm using because I'm using the latest version (I believe, 0.3.0) but in the previous question linked above, they were using 2.4 and it worked out for them.  I would really like to find a way to add +/- without having to manually add it with Tikz or something, i.e. just have it naturally part of the component without having to add nodes or extra Tikz code etc. If that's not possible, then I'll have to settle for that as a last result.   Thank you for your help!  EDIT: I should also mention that I've tried using battery1, v=$9V$ instead of battery1=\SI{9}{V} and to no avail (the +/- signs still don't show up). Execute script after specific delay using JavaScript Is there any JavaScript method similar to the jQuery delay() or wait() (to delay the execution of a script for a specific amount of time)? Can NDSolve handle discontinuous data? What's wrong in taking the cat as an observer in Schrodinger's experiment? Plz kindly elaborate! And if possible also describe about possible logics if the question bears the answer No. Order Bars in ggplot2 bar graph I'm rewiring a a small part of a wall and I'm confused about a combination of the NEC and the products that I'm able to find.  I'm working on a 20 amp circuit (with 12g wire of course).  I'm connecting 4 receptacles each in their own box, 15 amp receptacles for normalcy.  But the terminals don't seem to accept 12g wire, even on the screws.  My first reaction was to make pigtails 14g pigtails, so that the 14g tap goes directly to the receptacle and is not used to feed the rest of the circuit.  12g in the box, 12g out of the box, and 14g inside the box only.    Is that type of splice allowed, or should I look harder for a 15 amp outlet that can more easily accept 12g wire? Upvoting a downvoted answer will do +2 upvote? I have first downvoted an answer. Then, I decided to leave it the way it was before, meaning to undo my downvote just like it never existed, so I upvote to remove my downvote. Suprise! SO seems to consider it as two upvotes!  Let's say I have an answer with -3, then I decide to downvote it making it -4. After deeper thoughts, I prefer to remove my downvote to let it the way it was before I downvoted, meaning -3, but SO makes it -2 automatically!     Wouldn't it be supposed -3, and not -2? If someone else would have upvoted, it would have made it an upvote +1, not a +2!   Am I missing something here?  In fact, is there a way to tell SO that I only wish to undo my downvote, and not upvote the answer? I may understand why it does behave this way. I only wish to undo my downvote, though. I have for quite some time now tried to find a satisfactory answer to this, but I haven't yet. In perturbation theory, with small parameter $\lambda$, we expand the eigenstate as  $$| E \rangle=| E^{(0)} \rangle + \lambda | b \rangle + ...$$  Where $| E^{(0)} \rangle$ is an eigenstate of the unperturbed Hamiltonian. The problem is that when there is degeneracy, there is a choice of these eigenstates. I know the answer is that they are chosen to be eigenvectors of the perturbing Hamiltonian, but my question is why. Checking if file is checked out through CSOM Is there a way to check if a file is checked out by another user? I am going trhough a list file by file and would like to avoid the "file is opened by another user" error. Something like this would be nice:  Guid id = new Guid(slideLibrary); // ID of Slide-Library. List list = ctx.Web.Lists.GetById(id); ListItemCollection items = list.GetItems(CamlQuery.CreateAllItemsQuery()); ctx.Load(items); // Loading all the fields System.Security.SecureString pwd = new System.Security.SecureString(); foreach (char c in "pwd".ToCharArray()) pwd.AppendChar(c);  ctx.Credentials = new SharePointOnlineCredentials(login, pwd); ctx.ExecuteQuery();  // Iterating through all the entries. foreach (var item in items) {     String currentID = item["ID"].ToString(); // Getting the ID of the current item.      if (!item.IsCheckedOut()) // This is the part I am looking for.     {        // Do stuff     }     else     {        Console.WriteLine("Item is blocked, skipping it.")     } }   I have yet not been able to find something fitting. What is the amplitude of a light wave? Referring to this question   I'm curious about what is a amplitude of a light wave. Especially for light from a thermic source. How do I send SMS from PC through Android Phone or to Android Phone? I'm looking for an app that will let me type text messages on my PC and send the SMSs directly from the PC through my phone, or save the messages to the phone for later use.  I'm basically looking at something like Nokia PC Suite's equivalent.  EDIT:  Google Voice is only for US, doesn't work in India at least, and also defeats the purpose (it will go through the internet, why, when it's possible through the usb cable/modem/wifi). DesktopSMS is a good program, but kept crashing, for some reason. PDANet worked for me. But I liked the tiny . It's weird, and I had to change the bluecove library jar file for Windows 7 x64, but it's compact (a concern for me).