Proving $\sin^2x+\cos^2x=1$ I'm looking for a way to prove $\sin^2x+\cos^2x=1$ without using the Pythagorean Theorem or proving it.  The only thing I've tried was to play with the Taylor series, but I didn't get far with that as nothing seemed to 'mathify' nicely after a point.
Prove $\sin^2\theta + \cos^2\theta = 1$ How do you prove the following trigonometric identity: $$ \sin^2\theta+\cos^2\theta=1$$  I'm curious to know of the different ways of proving this depending on different characterizations of sine and cosine.
If $A$ is an $n \times n$ matrix and $ A^2 = 0$, then $\text{rank}(A)\le n/2$. Was revising for a Linear Algebra when I came across this question.     "Given a Matrix $A$ over $\mathbb{R}^{n \times n}$, and $A^2 = 0$,   show that rank($A$) $\leq n/2$"   My attempt:  I was thinking of the theorem from Cayley-Hamilton but the matrix is a $n \times n$ matrix hence that's probably invalid.  Is there perhaps a significance of $A^2 = 0$? apart from a special case of the nilpotent matrix? D'Alemberts Principle to solve a Classical Mechanics problem I found this question on this very site and was curious on how to solve it using D'Alembert's Principle  .I already know how to do it by balancing the torque about toppling point so please don't post the latter as an answer . My attempt was to displace the upper sphere by $d\theta$ which gives its virtual displacement as  $rd{\theta}cos(\theta)$j(notice we are only interested in the y component becuz the only component of the applied force which is gravity in this case is the y component in the dot product the other components vanish) similarly the cylinder is also displaced $r'd{\phi}cos(\phi)$ where r' is the distance from the point of toppling of the center of mass. $r'cos(\phi)=H/2=r(1+sin(\theta))$ ,$r'sin(\phi)=R=r(1+cos(\theta))$ and $tan(\phi)=H/2R$ the equation coming from D'Alemberts principle is $mgrcos(\theta)d\theta=Mgr'cos(\phi)\phi$ I tried solving this but couldn't arrive at the right answer. Any help would be appreciated! How do I split a string by a multi-character delimiter in C#? I’ve recently purchased a new Mac Mini, and I’ve been importing my old pictures into the Photos app.  I want to have these photos backed up to iCloud, along with the photos currently on my iPad. I then want everything synced to my Mac Mini, but not to my iPad, which has limited storage capacity.  Is it possible to have the Mac Mini and iPad set up to backup to iCloud, but only download backed-up photos to the desktop? How do users navigate to Meta? How do you expect users to:   Know that a site's meta-site exists (and what it's for)? Find the hyperlink to the meta-site?   I know it exists, of course, and how to get there; I'm wondering how (or even whether) you expect new or naive users to do that.  I think there isn't a link to Meta from the main site, am I right (except one link in the "list of all 171 Stack Exchange sites" drop-down menu on the toolbar)?  Sometimes I tell people to post a question on Meta instead of on the main site -- I don't realise that they may not know that Meta exists, and that if they do (because they've been there before) they don't know how to get there (because they haven't yet investigated everything on the toolbar). What are the details surrounding the brass brazier required for the Find Familiar spell? In our campaign, I play a warlock and brass braziers are hard to come by. I want to carry a brass brazier in case my familiar dies and I need to again cast find familiar.   How bulky is a small brass brazier? Can it be carried for several weeks without hindering long distance foot travel? That is, because of its shape or weight, does the typical brass brazier reduce my speed? What's the price of a small, simple brass brazier? How long would it take a brass worker to make a small, simple brazier? How long would it take the brass worker to make one specifically designed to be eminently portable? In both cases, assume the brass worker works nonstop exclusively on the brazier.     What I'm looking for are references, similar cases, and information that can help handle these issues properly and accurately. First and foremost in D&amp;D 5e (especially RAW) and then other RPGs and even real world, if they could be used to make estimates about the answers to these questions when we compare how similar cases translate to D&amp;D 5e mechanics. Is there ever a reason (aside from Outbound Messages) to use Workflow instead of Process Builder? Specifically a side-by-side comparison of how well each performs for single-condition field updates, for email alerts or for task creation.  This is not a question about differences (that's well documented) but whether there is any advantage to using Workflow at all.  Are there known obstacles that only exist in one but not the other that would help people decide when to use one or the other? Could be aspects like:   Resource Consumption  Time-based Behavior  Edge case options (e.g.selection of recipients for email) Deployment considerations Prove $\gcd(a+b,a^2+b^2)$ is $1$ or $2$ if $\gcd(a,b) = 1$ Assuming that $\gcd(a,b) = 1$, prove that $\gcd(a+b,a^2+b^2) = 1$ or $2$.   I tried this problem and ended up with $$d\mid 2a^2,\quad d\mid 2b^2$$ where $d = \gcd(a+b,a^2+b^2)$, but then I am stuck; by these two conclusions how can I conclude $d=1$ or $2$?  And also is there any other way of proving this result? PHP MySQLi doesn't recognize login info Android app chokes on per-site meta profiles Steps to reproduce:    Go to a per-site meta. Find any post. Touch the user that posted it. See something like this (click to enlarge):           As you can see, it says "Stack Overflow" as the site, and shows my SO rep and badges, but it shows my Meta.Code Golf posts. How to add readable text to a light and dark image? I'm trying to add text to a photo of a city but the colors in the background make it difficult to read light and dark text. I've tried several light and dark colors but they all looked overpowered by the colors in the image. What colors could I use or what can I do to the text to make it show but also make it so you can still see the background?  Here is the image I'm trying to use: I try to launch Firefox over SSH, using  ssh -X user@hostname   and then  firefox -no-remote   but it's very very slow.  How can I fix this? Is it a connection problem? What happens if my flight out of the Schengen area is cancelled on the last day of my visa? When you send a group text message in iOS, it appears as if you automatically opt in to receive any and all replies to that message. Is there a way to avoid being included in group replies? Is there a way to remove oneself from a group chat? After using the simple LM317 on breadboard variable power supply for a while, a need for a more rugged and reliable bench power supply has arose.    Short on money, i decided to build it as many other amateurs like me did. The thing is that, for my current needs i don't need a giant digitally controlled multi-channel 500 watt beast or something like that.  Instead what i really want is a low wattage positive and negative low voltage ouput -12v-0v-12v for example. However i really like that my power supply to be safe, reliable and fool-proof as far as possible. So after searching the web here and there, i found many designs that from my point of view fell in one of two divisions:-   Ones that use discrete op-amps and transistors. And ones that use integrated voltage regulators.   So -at least for me- i think that the ones with discrete op-amps and transistors are more prone to malfunctions and issues compared to the integrated ones with all of their internal over current and thermal protection circuits which would need to be discretely implemented in the bench power supplies that utilize discrete op-amps, however those designs have the advantage of ease of customization and more importantly that it is easier to have 0V output compared to 1.2V in integrated voltage regulator ones.    So i like advice to help me choose between circuits that utilize discrete op-amps like this circuit for example:-  Or ones that use integrated voltage regulators with their intrinsic robustness and low form factor.    Also as a side but related question i have seen many designs on the web that utilize negative voltage fed to the adjust or like pins to achieve 0V output from integrated voltage regulators like this single positive power supply:-     Or a positive and negative channels power supply that i really like its design because of simplicity (not because of laziness but to avoid the drawbacks that may come with complex designs especially when implemented by an amateur like me):- And the question is      Is it wrong or a system safety and reliability lowering act to feed negative voltage to adjust pin for LM317 for example?     I ask this question because the datasheet clearly states that the working output range is 1.2V-37V for LM317 for example, also no similar circuit is mentioned in the Typical Applications section except for the following one:- And even in that circuit, however the adjust pin is tied to the negative rail, it does't regulate during this interval instead it is shut until the next half wave at least that what i understand.    So at last to sum it up which alternative is best suited for my needs, its pros and cons and suggested circuits if possible. Does the new guidance on duplicate questions suggest closing a question as duplicate, if the original answers the OP's question? In other words, should we now look at the content of the answers when we are evaluating questions for possible duplication?  Is it sufficient to say "Your answer can be found in this question over here, therefore your question is a duplicate?"  The new guidance is here:    Among other things, it describes the new close banner:     This question already has an answer here:   Link to the Duplicate Question   ... which seems to suggest that, if some other question in the system answers the OP's question, then the OP's question can be closed as a duplicate on that basis (even if the question itself is not necessarily duplicated).  Example Here:    Note the comments below the question, where I dispute the assertion made there that it is a duplicate question.  The original question doesn't mention unit testing. $\int_0^\infty \frac{\ln\left(1+x-\sqrt{2x}\right)}{1+x^2}\,dx$ Trying to compute   I was facing: \begin{align}J=\int_0^\infty \frac{\ln\left(1+x-\sqrt{2x}\right)}{1+x^2}\,dx\end{align}  I want to prove that $\displaystyle J=0$, or equivalently, that,  \begin{align}\int_0^1 \frac{\ln\left(1+x-\sqrt{2x}\right)}{1+x^2}\,dx=-\dfrac{1}{2}\text{G}\end{align}  $\text{G}$ being the Catalan constant.  Read carefully please.   I know, using so-called Feynman's trick, how to prove this. I would like to obtain a proof, using only integration by parts and change of variable in simple integrals (that is, no multiple integrals) I don't know, if, under these restrictions, such computation is possible.  NB: You're probably wondering what is the link between : $\int_{0}^{\frac{\pi}4} \ln(\sin{x}+\cos{x}+\sqrt{\sin{2x}})dx$ and $J$.  The link is, for $x\in\left[0;\frac{\pi}{2}\right]$, $(\sin{x}+\cos{x}+\sqrt{\sin{2x}})(\sin{x}+\cos{x}-\sqrt{\sin{2x}})=1$ and $\sin(2x)=2\sin x\cos x$ Does Professor X have any responsibility for the state of the world? Throughout the course of Logan, it's obvious that Professor Xavier is having issues controlling his powers, and at the end of the movie, just before      X-24 kills him   he claims to "remember" what he did and expresses regret. We also hear about an incident in Westchester that killed several mutants, that sounds like Xavier may have been involved. Is Xaiver the reason there are no more mutants? Is that why he's sorry, because he knows he killed them all? Guaranteed lifetime of temporary in C++? How To Show Node Comment Count in node.tpl? I have had the bright idea of changing my user permissions so as guests cannot see the comments on a node. I have no problems with Views around my site, as they can clearly see there are comments on nodes, and they will be more inclined to register. The problem seems to be when they arrive on my site directly to a node, they don't see any comment, and think "that's a bit crap; let's go."  Is there a way I could show the amount of comments below a node?