Proof: $\cot(7.5^\circ) = \sqrt2 +\sqrt3 + \sqrt 4 + \sqrt6$ I am working on this problem, but I am not able to simplify the expression.   So please help me.  Thanks in advance
Prove that: $ \cot7\frac12 ^\circ = \sqrt2 + \sqrt3 + \sqrt4 + \sqrt6$ How to prove the following trignometric identity? $$ \cot7\frac12 ^\circ = \sqrt2 + \sqrt3 + \sqrt4 + \sqrt6$$  Using half angle formulas, I am getting a number for $\cot7\frac12 ^\circ $, but I don't know how to show it to equal the number $\sqrt2 + \sqrt3 + \sqrt4 + \sqrt6$.  I would however like to learn the technique of dealing with surds such as these, especially in trignometric problems as I have a lot of similar problems and I don't have a clue as to how to deal with those.  Hints please!  EDIT:  What I have done using half angles is this: (and please note, for convenience, I am dropping the degree symbols. The angles here are in degrees however).  I know that $$ \cos 15 = \dfrac{\sqrt3+1}{2\sqrt2}$$  So,   $$\sin7.5 = \sqrt{\dfrac{1-\cos 15} {2}}$$ $$\cos7.5 = \sqrt{\dfrac{1+\cos 15} {2}} $$  $$\implies \cot 7.5 = \sqrt{\dfrac{2\sqrt2 + \sqrt3 + 1} {2\sqrt2 - \sqrt3 + 1}} $$
What exactly happens when a Weeping Angel is quantum locked? Interpretation of p-value in Mann-Whitney rank test I am testing Mann-Whitney rank test with two vectors a and b. Vectors are almost similar so I expect a p-value near 0, but the returned p-value is near 1. What is the reason? I read  and also run the code with different parameters, but don't get anything near to what I expect.  from scipy.stats import mannwhitneyu import operator import numpy as np  a = [1000,100,10,1,10,100,1000,10000,1000,100,10,1] b = [999,100,10,1,10,100,1000,10000,1000,100,10,1]  print(mannwhitneyu(a, b))   The output of the code:  MannwhitneyuResult(statistic=73.5, pvalue=0.95291544831453867)   Edit:  Let's formalize the problem to eliminate any misunderstanding (please edit the question if something is wrong):  What I try to prove is that rank distribution of data are approximately equal,   Null hypothesis ($H_0$)= "ranked distribution of a and b are approximately equal"  Alternative hypothesis ($H_a$)= "ranked distribution of a and b are not equal"  significance level (alpha) = 0.05  p-value= 0.95  (p-value > alpha) so there is no sufficient evidence that $H_a$ is correct, but we cannot also conclude that ($H_0$) is true.   Instead if (p-value &lt; alpha) was true, then we would have enough evidence against H0 and Ha can be accepted.  So what happened here is that I could not disprove the null hypothesis. However, it doesn't mean that null hypothesis is false. It is like an investigation to accuse MR X of being guilty:   $H_0$=" MR X guilty" $H_a$=" MR X not guilty"  We guess he is not guilty, but we don't have enough proof against him: (p-value > alpha), but that doesn't mean that ($H_0$) would not be true. If we could obtain enough evidence and state (p-value &lt; alpha), then we disprove $H_0$ and we can conclude that Ha is true and he is not guilty. Conclusively stop wake timers from waking Windows 10 desktop How do I delete multiple rows in Entity Framework (without foreach) Stupidly, I had been using a condition like this as part of a script:  if [ $(ls FOO* 2&gt; /dev/null) ] # if files named "FOO*" were downloaded  then     echo "Files found"      # ... process and email results else     echo "Not found"     # ... email warning that no files were found (against expectations) fi   That works for zero and one files named FOO*, but fails if there are more than one. From logs I found several different error messages stemming from this:  [: FOO_20131107_082920: unary operator expected [: FOO_20131108_070203: binary operator expected [: too many arguments   My question is: what is the correct way to check, in a Bash if condition, whether one or more files whose name begins with FOO exist?  GNU bash, version 4.2.25(1)-release (x86_64-pc-linux-gnu) How can I set an external display (monitor) as the primary one? Currently the MacBook's display works as primary, so every application I open gets open on the 13" and I have to drag it to the 24". I also don't have access to the dock on my 24".  Is is possible to turn off the MacBook screen and use only the one attached via DVI?  (I've just connected my 24" HP LP2475w via Mini DisplayPort to DVI to my MacBook Pro 13", which works great except for this one thing.) Is it acceptable to write a thank you in a comment? There have been a number of cases where I would like to thank someone for providing me with a thorough response and give them feedback beyond the vote up. Often I wish people would tell me why they voted me up. Not that the thank you part is necessary but it sets it apart from the other comments related to answering the question itself.  In this type of situation is a "Thank you" acceptable in a comment?     For more information, see  in the . How to disable automount in nautilus's preferences Cyclic Quadrilateral Problems Let ABCD be a cyclic quadrilateral. The diagonals AC and BD meet at P, and DA and CB produced meet at Q. The midpoint of AB is E. Prove that if PQ is perpendicular to AC, then PE is perpendicular to BC. Can the Hubble constant be measured locally? Calling acro first use when plural is producing an 's' in the short definition that is wrong I have an acro first use that must be plural, but I am getting an erroneous "s" when I shouldn't be. I'm getting (sIMFs) when it should be (IMFs). See picture and code below. The problem only occurs on the first call and it does so for all my acronyms. Thanks in advance.   \documentclass[12pt,twoside]{report} \usepackage{acro} \DeclareAcronym{IMF}{     short = IMF,     long  = Intrinsic Mode Function,     class = abbrev}  \begin{document} \acp{IMF} \acresetall \ac{IMF} \acp{IMF} \acfp{IMF} \end{document} Write this equation I want to write this equation in this way down but I have been unsuccessful. How can I improve this codes in order to get the equation below?  \[ DeltaC{t} = \left\{ \begin{array}{ll}  -1.40- 0.588w_{t-1} - 0.5351\DeltaC_{t-1}+ 0.169\DeltaBC_{t-1} &amp; \mbox{ if $w_{t-1}&lt;- 0.62$} \\ -1.40 - 0.588w_{t-1} - 0.5351\DeltaC_{t-1} + 0.169\DeltaBC_{t-1} &amp; \mbox{$w_{t-1}&gt;0.62$} \end{array} \right. \] pyvenv vs venv vs python-virtualenv vs virtualenv and python 3 malikarumi@Tetouan2:~$ pip install virtualenv     Collecting virtualenv     Downloading virtualenv-12.0.7-py2.py3-none-any.whl (1.8MB)     100% |################################| 1.8MB 330kB/s   malikarumi@Tetouan2:~$ pip freeze     (a lot of stuff, but not virtualenv)  malikarumi@Tetouan2:~$ virtualenv testvenv1     The program 'virtualenv' is currently not installed. You can install it by typing:     sudo apt-get install python-virtualenv   What is going on here? is python-virtualenv == pyvenv? Isn't that still broken? Does original virtualenv still work with python? if venv (too many name variations!!!) is part of the standard library, , why am I being told to install it?  And when I did try to install it, I got:  malikarumi@Tetouan2:~$ sudo apt-get install python-virtualenv     Reading package lists... Done     Building dependency tree            Reading state information... Done     The following extra packages will be installed:     python-colorama python-distlib python-html5lib python-pip python-setuptools   At which point I aborted, because installing python3-pip just before that had given me   Setting up python3-setuptools (3.3-1ubuntu1) ... Setting up python3-pip (1.5.4-1) ...   And I wasn't sure if the extra packages would overwrite them or go onto python 2.7.  I read that python 3 is supposed to be the default going forward. Since both 2.7 and 3.4 are there, and have separate commands, how do I not only make sure Python 3 is my default, but that anything I install goes there and is used by it instead of 2.7?   BTW, I tried sudo apt-get python3-virtualenv and got: E: Unable to locate package python3-virtualenv I am using the Service Console in Lightning Experience, and using the Chatter Feed component on the Case layout to display updates to the Case.  If I use the "Log a Call" Quick Action from the Chatter publisher, these calls are displayed in the Chatter feed.  This is the desired behaviour.  However, if I insert a Log a Call record related to the Case any other way - e.g. using the Developer Console, or my CTI system's custom components - the call does not display in the feed.  If I query the record, it looks exactly the same as the record that is created with the Quick Action.  It just seems that the Chatter post is never made in this situation.  I have checked my Feed Tracking settings:   Task has Enable Feed Tracking and Subject checked Case has All Related Objects checked   How can I enable calls created with a custom component (part of a CTI system) to display in the case feed? Terminal limits scrolling I have a running log, it outputs a lot to the terminal. Unfortunately, when I try to scroll up and view older output, it appears there is a limit. And I am unable to scroll past a certain point. On the Mac, the terminal allows you to scroll up to the start of the session. Is there a way to do this in ubuntu terminal? Clearing all cookies with JavaScript Prove that $z^2 \equiv ab\pmod p$ is solvable if and only if both or neither of $x^2 \equiv a \pmod p$, $x^2 \equiv b \pmod p$ are solvable. Suppose the $p$ is an odd prime not dividing $ab$.  Prove that $z^2 \equiv ab \pmod p$ is solvable if and only if both or neither of $x^2\equiv a\pmod{p}$ and $x^2 \equiv b \pmod p$ are solvable. What is JavaScript's highest integer value that a number can go to without losing precision? Is this defined by the language? Is there a defined maximum? Is it different in different browsers? When I use ssh -X on my Mac (running OS X 10.6.7) to connect to my Ubuntu (11.04) box, I get the following warning:     Warning: untrusted X11 forwarding   setup failed: xauth key data not   generated Warning: No xauth data;   using fake authentication data for X11   forwarding.   Is there something I can do to make this warning go away?  If not, can I safely ignore it?  X11 forwarding seems to work fine, though I do see this message:     Xlib:  extension "RANDR" missing on   display "localhost:10.0".   Is that related to the warning?  (I'm guessing not.  If it's not, I'll file a new question about that.) My processor is 64-bit - does that mean I need the amd64 image? My processor is an Intel Core 2 Duo P8600 (2.40GHz). As far as I know that's a 64-bit processor - I'm a bit confused as the architecture is called AMD 64, is this a generic name given to 64-bit architectures? I've heard of x64 but can't see a release labelled with this.