I am just curious to know the direct and precise to prove the identities  (1) $$\cos(A+B)=\cos A \cos B-\sin A \sin B$$ and   (2)  $$\sin(A+B)=\sin A \cos B+ \cos A \sin B$$ I tried to come up with right-angled triangles but I failed to connect them and get some logic out. Thank's in advance.
The "" formulas often come in handy, but it's not immediately obvious that they would be true.  \begin{align} \sin(\alpha \pm \beta) &amp;= \sin \alpha \cos \beta \pm \cos \alpha \sin \beta \\ \cos(\alpha \pm \beta) &amp;= \cos \alpha \cos \beta \mp \sin \alpha \sin \beta \end{align}  So what I want to know is,    How can I prove that these formulas are correct? More importantly, how can I understand these formulas intuitively?   Ideally, I'm looking for answers that make no reference to Calculus, or to , although such answers are still encouraged, for completeness.
I want to know where the Metro applications (Mail, People, Messages, ...) executable file path is.  Also, how can I run Mail application when my PC starts? GNU screen: Running commands in different window when screen is started I am attempting to search jobs by tag on StackOverflow Careers. On SO, this is done using square brackets, but that is simply doing a keyword search, and I still get results that don't have the tag I am looking for.  For example, when I search for the keyword "android", I get this:        which is the same as when I search for the keyword "[android]":        Is there a format I can use to search jobs by tag? If not, I believe this would be a nice feature (seems intuitive for the Stack Exchange invite-only job site). I have an external hard drive and I did something which I shouldn't have done, and I am annoyed that many files are prompting me about Administrator permissions to move a folder.  They are regular files.  How do I reset all the permission on the external hard drive? Nonabelian semidirect products of order $pq$? I just constructed the semidirect product in Lang, and I'm trying to tie some facts together. From Ash's Algebra, I know that if $p\lt q$ are distinct primes, if $q\not\equiv 1\pmod{p}$, then any group $G$ of order $pq$ is abelian.  Is the converse true, that for any primes $p\lt q$, if $q\equiv 1\pmod{p}$ then there exists a nonabelian group of order $pq$?  One example I found online is that $\mathbb{Z}_3\ltimes \mathbb{Z}_7$ is nonabelian, and here $7\equiv 1\pmod{3}$. I was considering then semidirect products $\mathbb{Z}_p\ltimes\mathbb{Z}_q$ where $q\equiv 1\pmod{p}$ and some homomorphism $\phi\colon \mathbb{Z}_p\to\operatorname{Aut}(\mathbb{Z}_q)$ I calculate that  $$ (1,0)(0,1)=(1+0,\phi_0(0)+1)=(1,1) $$ and $$ (0,1)(1,0)=(0+1,\phi_{-1}(1)+0)=(1,\phi_{p-1}(1)). $$ Is it true somehow that $\phi_{p-1}(1)\neq 1$ in each case to show the group is nonabelian? I guess if it did this would imply $\phi_{p-1}$ is the trivial automorphism, so maybe there's something there? If not, is there a way to show $\mathbb{Z}_p\ltimes\mathbb{Z}_q$ is nonabelian in these cases in general? Thanks. UV coordinates on a deformed plane I have a polygon shape file that has many overlapping features. Each feature represents an area used by polar bears. So some of the polygons overlap and some don't, and there are separate clusters of overlapping areas.   I would like to get a measure of how much overlap there is in a particular area. But I have no clue how to do this. I'm pretty new to GIS and am still learning the ropes, it just seems I can't find an answer. Evalute $ \lim_{n\rightarrow \infty}\sum^{n}_{k=0}\frac{\binom{n}{k}}{n^k(k+3)} $ Why are old initrd files of uninstalled kernels filling up /boot partition? I am struggling to understand this. According to truth tables, if $P$ is false, it doesn't matter whether $Q$ is true or not: Either way, $P \implies Q$ is true.  Usually when I see examples of this people make up some crazy premise for $P$ as a way of showing that $Q$ can be true or false when $P$ is something outrageous and obviously untrue, such as "If the moon is made of bacon-wrapped apple-monkey carburetors, then I am a better wakeborder than Gauss."  $P$ is clearly false, but $P \implies Q$ is true no matter what the state of $Q$ is, and I don't understand why.  Are we saying "If $P$ is false, then all bets are off and $Q$ can be anything, either true or false, and not contradict our earlier claim, and if it isn't false, it must be true"?  Otherwise why can't we say that if $P$ is false, then we can't make any claims one way or the other on whether or not it implies anything at all? Minimum eigenvalue and singular value of a square matrix How to show that the relationship $\left | \lambda_{min} \right | \geq \sigma_{min}$ holds between the minimum eigenvalue and singular value of a square matrix $A \in \mathbb{C}^{n \times n}$? How to show that the set of orthogonal n x n matrices forms a group under multiplication I am studying orthogonal matrices and I am not sure if to show if a set of orthogonal $n \times n$ matrices forms a group under multiplication. We must check each of the group axioms. I found that the axioms are:  Closure Associativity Existence of identity matrix Existence of the inverse matrix.  I edited my question, since I was able to find more information about this topic. This group is called $O(n)$. To check the four axioms I did: Let $A \text{ and } B \in O(n)$, denoted as orthogonal matrices and assume that $C=AB$, then, Closure :  To prove that $C \in  O(n)$ we must prove that $C$ is a real $n \times n$ orthogonal matrix with uni-modular determinant. Since A and B are real $n \times n$ matrices, $C$ is also a real $n \times n$ matrix so, $C^TC=(AB)^T AB=B^T A^T AB = B^TB=I$  Associativity :  Matrix multiplication is associative, so the law holds for $O(n)$ group elements.  I am not sure if this is enough to prove associativity. Identity element :  The $n \times n$ identity matrix $I_{n \times n}$ represents the identity element.  In this case I am not sure if this is enough to prove the identity element. Inverse element :  Let $A^{-1}$ be the inverse of $A$, then we need to prove that  $A^{-1} \in  O(n)$ since $(A^{-1})^T=(A^T)^{-1}$. We have that: $(A^{-1})^T A^{-1}=(A^T)^{-1} A ^{-1}=(AA^T)^{-1}=I^{-1}=I$  Can anyone check if what I did is correct? I also would like to know if I can prove the associativity and the identity element in a better way. thanks Do redirection operators always open file descriptors in parallel? 1. Consider snippet#1:  $ cat test.txt &gt; test.txt cat: test.txt: input file is output file   It seems that cat makes its input file descriptor point to test.txt and then when it tries to set its output file descriptor to test.txt it throws the above error. Here it seems that cat is aware of the redirect operator and so proceeds to attempt to set the output file descriptor to test.txt  2. Consider snippet#2:  $ cat 1.txt 1:CAT 2:dog $ sed 's/cat/CAT/g' test.txt 1:CAT 2:dog $ sed 's/cat/CAT/g' test.txt &gt; test.txt $ cat test.txt # Note that test.txt is now empty $   Here we see that sed opens test.txt (last argument) in read mode and at the same time sets test.txt as its file output descriptor. Also the '&gt;' operator overwrites the contents of the file BEFORE sed starts to read from it.   I am aware that commands in a pipeline execute in parallel but have not come across any info on how redirection operators behave. Any supporting links would be helpful. These should work by adding spaces or removing spaces from the beginning of the current line or every line between the start and end of a text selection (inclusive).  (edited this question to supplement ) Is this arthropod non-intrusive and recommended to be left alive? Atheros AR9462 WiFI very unstable & package loss I'm having severe issues w/ package-loss and a generally bad connection w/in a WiFi-network at my uni. I'm usually encountering a package loss of ~50% and aborting connections which makes working w/ SaaS impossible.  I'm using Ubuntu 13.04 and my network-adapter is an Atheros AR9462.  Applying the nohwcrypt=1-option doesn't change anything.  Also very noticable: The connection works as intended in other networks; the problematic network uses WPA2 w/ MSCHAPv2.  Anyone can help my out here?  $ sudo lshw -class network  *-network UNCLAIMED             description: Ethernet controller        product: NetLink BCM57780 Gigabit Ethernet PCIe        vendor: Broadcom Corporation        physical id: 0        bus info: pci@0000:04:00.0        version: 01        width: 64 bits        clock: 33MHz        capabilities: pm msi pciexpress bus_master cap_list        configuration: latency=0        resources: memory:d3500000-d350ffff   *-network        description: Wireless interface        product: AR9462 Wireless Network Adapter        vendor: Atheros Communications Inc.        physical id: 0        bus info: pci@0000:09:00.0        logical name: wlan0        version: 01        serial: e0:06:e6:76:ba:89        width: 64 bits        clock: 33MHz        capabilities: pm msi pciexpress bus_master cap_list rom ethernet physical wireless        configuration: broadcast=yes driver=ath9k driverversion=3.8.0-22-generic firmware=N/A ip=131.173.74.130 latency=0 link=no multicast=yes wireless=IEEE 802.11abgn        resources: irq:19 memory:d3400000-d347ffff memory:9fb00000-9fb0ffff Paper status changed to “Decision in Process” rapidly after initial submission How can I secure an Ethernet-over-power network? I just bought myself an Ethernet-over-power adapter, and plugged it in. It seems to be working great, but now I'm concerned that my neighbours may be able to access my network.  I live in a rowhome complex; is it possible they can access my network? If so, how can I secure it? Restrictions on the variables for this identity are as follows: $a \geq m \geq 0$ and $n \geq b \geq 0$.  I have not gotten far with this, but these are my very rudimentary thoughts: We need to define a special element, somehow, and then carve up the $m + n$ remaining elements to be chosen out of the remaining pool of $a + b$ elements by writing $a + b = a - i + b + i$. From the $a - i$ pile, we choose $m$ things. From the $b + i$ pile, we choose $n$ things. My problem is, what is going on with this special element? How is it being defined so as to split things in this precise way?  A hint would be great for the moment. I accidentally opened Orca Screen Reader from the Dash when I was trying to open Screenshot. Now everything I type or click on is spoken out loud.  How do I make it stop?  For reference, here is a screenshot of Orca Screen Reader running: