If $\tan\left(\frac{\pi}{4} + \frac{y}{2}\right) = \tan^3\left(\frac{\pi}{4} + \frac{x}{2}\right)$, prove that $\sin(y) = \sin(x)\left[\frac{3+\sin^2(x)}{1 + 3\sin^2(x)}\right]$  I come until the step where  $ \cos{y}=\cos^3{x}$  After this I dont now what to do
If $$\tan(\frac{\pi}{4}+\frac{\theta}{2})=\tan^3(\frac{\pi}{4}+\frac{\alpha}{2})$$, then prove that $$\sin \theta=\frac{3 \sin \alpha+\sin^3 \alpha}{1+3\sin^2 \alpha}$$  I tried using the fact that  $\frac{\cos A}{1-\sin A}=\tan(\frac{\pi}{4}+\frac{A}{2})$ but now not able to eliminate $\cos \alpha$ and $\cos \theta$. How should I proceed?
Why are Only Real Things Measurable? Why can't we measure imaginary numbers? I mean, we can take the projection of a complex wave to be the "viewable" part, so why are imaginary numbers given this immeasurable descriptor? Namely with operators in quantum mechanics, why must measurable quantities be Hermitian, and consequently real? Why does everything spin? The origin of spin is some what a puzzle to me, everything spin from galaxies to planets to weather to electrons.  Where has all the angular momentum come from? Why is it so natural?  I was also thinking do photons spin? we always think of the wave as a standard 2d sin wave but could this rotate in 3d? What implications would this have?   And what about spacetime how does all the spinning effect?  This has always been avoided in all lectures and classes I ever went to. Usage of: 'since', 'for' and 'in' Dimensional Tolerance for US Standard NEMA 1-15 Plugs I am trying to find the dimensional tolerance on US electrical plugs. I have tried looking online, but I have only been able to find that they are called , and have found the picture below showing length and height for the male connector (but not width of the male prong), and I haven't been able to find dimensional tolerances anywhere.     I ask here because I'm assuming there's a US code that governs electrical plugs, but I can't seem to find the standard anywhere. Retain precision with double in Java How do I make my document use the Times font, both for the text and the math? I know that I can put \usepackage{times} in the preamble in order to use the Times font instead of Computer Modern for my document.  But this only affects the text; the math is still typeset using Computer Modern.  How can I make Latex use Times also for the math? How to run long time process on Udev event? I want to run a ppp connection when my USB modem is connected, so I use this udev rule:  ACTION=="add", SUBSYSTEM=="tty", ATTRS{idVendor}=="16d8",\     RUN+="/usr/local/bin/newPPP.sh $env{DEVNAME}"  (My modem appears in /dev as ttyACM0)  newPPP.sh:  #!/bin/bash /usr/bin/pon prov $1 &gt;/dev/null 2&gt;&amp;1 &amp;   Problem:  The udev event fires, and newPPP.sh is running, but newPPP.sh process is killed after ~4-5s. ppp does not have time to connect (its timeout is 10s for dial up).  How can I run a long time process, that will not be killed?  I tried using nohup, but it didn't work either.  System: Arch Linux  Update  I found a solution , thanks to .  I use at now to run my job detached from udev process.  But the one question remains unanswered: Why do nohup and &amp; not work? If $\sqrt[3]{a} + \sqrt[3]{b}$ is rational then prove $\sqrt[3]{a}$ and $\sqrt[3]{b}$ are rational Assume there exist some rationals $a, b$ such that $\sqrt[3]{a}, \sqrt[3]{b}$ are irrationals, but:  $$\sqrt[3]{a} + \sqrt[3]{b} = \frac{m}{n}$$  for some integers $m, n$  $$\implies \left(\sqrt[3]{a} + \sqrt[3]{b}\right)^3 = \frac{m^3}{n^3}$$  $$\implies a + b + 3 \cdot \sqrt[3]{ab}\left(\sqrt[3]{a} + \sqrt[3]{b}\right) = \frac{m^3}{n^3}$$  Since $a +b$, $\sqrt[3]{a} + \sqrt[3]{b}$ are rational, $\sqrt[3]{ab}$ must be rational as well.   For convenience let us say $\sqrt[3]{a} = p, \sqrt[3]{b} = q \implies pq$ is rational. This means, for all $i$, $p^iq^i$ is rational.  $$\implies (p + q)^2 = \frac{m^2}{n^2}$$  $$ = p^2 + q^2 + 2pq = \frac{m^2}{n^2}$$  Since $pq$ is rational, $2pq$ is rational and so is $p^2 + q^2$.  Assume, for some $i$ that $p^i + q^i$ and $p^{i-1} + q^{i-1}$ is rational.   $$\implies (p^i + q^i)(p + q) - pq(p^{i - 1} + q^{i - 1}) = p^{i+1} + q^{i+1}$$  is rational as well.  So for all $i$,  $$a^{\frac{i}{3}} + b^{\frac{i}{3}}$$  and   $$a^{\frac{i}{3}}b^{\frac{i}{3}}$$  are rational.  I know I'm really close to the answer, but it somehow just keeps slipping through my fingers. Windows keeps making these random "device connect" and "device disconnect" sounds Throughout the day I notice that my computer makes apparently random device-connect and/or device-disconnect ("boink") sounds.    I suppose it is the same sound you hear when connecting or disconnecting a USB device such as a thumb drive.  I've noticed that this happens on each of three computers I work with at home, my wife's computer, and my machine at work.  It happens without any user action at all - i.e. I'll be just sitting there (hands off my mouse and keyboard), and the computer will make the sound.  There is no visual queue or anything.  Just the sound.  I have sometimes gone in pursuit of the sound - running virus scans, examining event logs and such, and observing task manager - but have never had any luck tracking this thing down, but have not had any luck.  I've been noticing this since Windows Vista and now the sounds have followed me through to Windows 7.   Surely someone else out there must be experiencing this and can explain what is happening. propositional calculus problem, how to prove this right or wrong? $A$$\rightarrow$$(B$ $\vee$ $C$ ) , $B$ $\rightarrow$ $C$ $\vDash$ $A$ $\rightarrow$ $D$  I think it's wrong but I have no idea how to prove. How to compile self-referencing functions (to perform recursive a task) Why is my command throwing a 'data tag parsing failed' error? I am trying to lock some boxes in a map of mine, and I used this command:   /blockdata 10 4 9 {Lock:Elyiana's Mage Key}   At first it worked, but then when I logged back on the next day it gave me this error message:    Data tag parsing failed: Expected '}' but ''' at: {Lock:Elyiana'&lt;--[HERE]     I don't know why it doesn't like the apostrophe in the key's name. I would change it, but it already worked and locked some other items. My version is 1.12, does anyone have an answer to my problem? Irreducible conics An algebraic set (not necessairly a variety) $X \subseteq \mathbb{A}^2$ defined by a polynomial of degree $2$ is called a conic.  The problem is:     Show that any irreducible conic is isomorphic either to $Z(y-x^2)$ or to $Z(xy-1)$ after an affine change of coordinates in $\mathbb{A}^2$.   However, with this definition of conic, I only can conclude that if the polynomial is irreducible, then the conic is irreducible (because $f$ irreducible $\Rightarrow Z(f)=X$ is irreducible)     What can I do to prove that if a conic is irreducible, then the polynomial that defines the conic is irreducible?   And for the main problem, how can I do the affine change of coordinates?  Thanks How to access a usb flash drive from the terminal? I only have access to the command line. I need to backup my data (on the user folder) to a pen (USB DOK).    How can I mount a flash drive manually? What should be the copy command? Number of bases of an n-dimensional vector space over q-element field. If I have an n-dimensional vector space over a field with q elements, how can I find the number of bases of this vector space? I have a laptop which is connected to the internet with ADSL modem, and I would like to share the internet for any laptop connecting to wireless router from my laptop using hotspot.  This is what I want:  ┌────────────────────┐ │ ADSL modem         │ └───┬────────────────┘     │ ┌───┴───────────────────────┐ │ laptop's wired adaptor    │ │                           │ │ Hotspot software          │ │                           │ │ laptop's wireless adaptor │ └───┬───────────────────────┘     │ ┌───┴──────────────────────┐ │ wireless router(linksys) │ └───┬──────────────────────┘     │ ┌───┴─────────────────────────────┐ │ Laptop computers and cellphones │ └─────────────────────────────────┘   How can I do that? If you look at this product:    you can clearly see that it is a DVD+R, i.e. once it is written, it cannot be erased/rewritten.  Then what does the "RW" logo (in the lower right corner) mean? I thought it stood for ReWritable, but experimenting with the medium shows that it is not. Data recovery from an accidental format on ext4 partition When I was installing Mint Debian edition unlike the classic edition, the installation automatically formated my home partition when I did not specify to format.  So the formatting previously was ext4 as is now. I believe the data is still there as it was a quick format.   I have now booted the computer up on a live USB to prevent writing on it.  Ran testDisk. Is there anyway to recover to a previous superblock so i can recover my data? Get terms that contain posts that in turn belong to other terms? So I have custom taxonomy called "shape" and another one called "color". If I want to get the list terms under shape that contain posts that are also present in terms under color, how do I do that?   Example, I have a post under custom post type toy called Ball which is present in both shape Circle and color Red. I want to retrieve the term name Circle via $wpdb query. I'd like to emphasize that I want to retrieve the list of terms and not the posts. How can I get suffix matching in a search in SharePoint 2013? I'm new to SharePoint 2013 and I'm quite puzzled, that it seems to be impossible to search for the middle part of a string.  Simple example:  I've added a lot of documents (PDFs, ...) from our Windows Server to the SharePoint Library and I want to add metadata. I try to filter the view of the document library, to only get those documents which I currently want to edit.  The names of the documents are very systematic, composed of naming conventions with abbreviations like:  Superthing2000-b_Suppl_REF_13580_note_1108_04en which means e. g.: Document about Product "Superthing2000", variant "b", Supplement to the manual, ID "13580", from 08/2011 in english language.  Now, I'd like for example to filter out all documents containing ID 13580 or "en" for english, but no chance, as suffix matching is not possible, even if I explicitly search for e. g. *en*.  Is there a way to define a search with kind of a "contains" operator which also finds files when the search string is only a part in the middle of the file name? Even the Windows Explorer search feature is able to do that!!