Prove that: $\frac {2sin x}{cos 3x}+ ...$
Show that $\frac{\sin x}{\cos 3x}+\frac{\sin 3x}{\cos 9x}+\frac{\sin 9x}{\cos 27x} = \frac{1}{2}\left(\tan 27x-\tan x\right)$
Does a correlation matrix of two variables always have the same eigenvectors? I perform Principal Component Analysis using two variables that are standardized. This is done by applying a SVD on the correlation matrix of the concerned variates. However, the SVD gives me the same eigenvector (weights) irrespective of what the two variables are. It's always [.70710678, .70710678]. I find this strange. Of course, the eigenvalues differ.  My question is: How to interpret this?    PS. I wanted to conduct a total least squares regression on two variables. My statistical programme does not provide TLS, but TLS luckily equals Principal Component Analysis, as far as I know. Hence my question. The question is not about TLS directly, but why I get the same eigenvectors irrespective of which variables I use (as long as they are exactly 2). Similar to Lifepod 1993, but not it How much to tip in the United States? In a multi server architecture, is it possible to find out from which machine a user logged into current machine using ssh?  For example, user1 logged into host1. From there, the user logged into host2 as sruser using ssh.  In above example, would it be possible to trace the sruser session of host2 back to user1 session in host1?  Would it be possible to get the hostname or terminal of user1 session? What are some examples of manifolds that do not have boundaries and are not boundaries of higher dimensional manifolds?  Is any $n$-dimensional closed manifold a boundary of some $(n+1)$-dimensional manifold? I started using Sublime Text 2 a week or so ago and I loved it so much I've spent $60 on it and now want to use it for everything.  Gedit currently has ownership of most of the text file associations. I'd essentially like to replace any association to gedit with sublime-text-2.  Any tips available on bulk association changing? Desktop-neutral methods are preferred. Should 'drive by' downvoting be more effectively caught? I have an answer that I'm certain is technically correct. However, it continues to be downvoted. The answer has already been noted in comments on other answers to be better. It goes more in-depth than the others, yet it continues to be downvoted.  The already upvoted answers were nowhere close initially, and I had already commented on them to say why it is wrong...  Isn't there a limit for that? Shouldn't something be catching this? Do I have to move swap partition to the right side? According to  I just expand my sda4 to include unallocated space, my question is after create 2 new partitions sda7, sda8, do I need to move the swap partition to the most right of the disk? (I really don't know this, I saw  which moving the swap partition to the right side)  Thanks. Are there published predictions that hydrogen could remain metallic at ambient pressure? Below the question  there is a  that directed me to the BBC News summary . In their attempt to balance the skepticism of this particular reported observation with the potential implications of producing metallic hydrogen in a laboratory, the BBC included some far-reaching possibilities:  If that is true - and it is a controversial claim - it fulfills a more than 80-year quest to produce what many have said would be a wonder material. Theory suggests metallic hydrogen could be used to make zero-resistance electrical wiring and super-powerful rocket fuel, among many applications.  At the end of the article:  &quot;The scepticism here is probably a good thing, in that it will drive many groups towards attempting to reproduce this experiment. This publication will certainly incite the field. Again, if it holds up, this is an exciting result. I think in this case time will tell,&quot; he told BBC News.  ...referring to Marcus Knudson from Sandia National Laboratories.  And Jeffrey McMahon from Washington State University concurred: &quot;With respect to the tiny sample amount: Such experiments are performed in small diamond anvil cells. One challenge would be to make a larger quantity (at once); another, perhaps bigger challenge is to recover even the small sample (ie, remove it from the extreme pressures that it is under in the diamond anvil cell). &quot;Whether the latter is possible is an important open question.&quot;  Question: Are there published, peer-reviewed predictions that hydrogen could remain metallic at ambient pressure? A link or reference would be appreciated. If it's still in pre-print form (AeXiv) that's OK too. If it's possible, a little help understanding the prediction would be great as well: I know there are many predicted crystalline forms of water ice at elevated pressures and/or reduced temperatures, and many or most of these have been observed, but I think most of them revert once pressure is reduced to ambient. Also the  comes to mind. If there is such a prediction of the possibility of hydrogen remaining metallic at ambient, is there an analogous example of this with a material I'm more familliar with, or is this something quite new? Add high-resolution-graphics and / or .svg-icons Refreshing open attribute table in ArcMap using ArcPy rather than ArcObjects? I am running the Calculate Field GP tool from the arcpy in ArcMap. When a row value is updated, it does not show up untill I use the Reload Cache tool (Table Options > Reload Cache). How do I refresh the attribute table automatically without using ArcObjects?  I have tried updating the definition query as @blah238 suggested , but this doesn't help. Resetting the data source is not an option because the feature class might be in the editing session and I don't want to lose the connection to the source data. I have a simple circuit:    The max rating (current) of the LED is 30mA.  How can I work out what the resistance of the resistor needs to be? By using ohm's law I found it to be \$3V/0.03A = 100 \Omega\$. However using software called , and trial and error I got the minimum possible resistance to be 36&Omega;. However, if I use a 35&Omega; resistor, then the LED breaks. Is the software wrong, or (more likely) am I doing something wrong? A circle has the same center as an ellipse and passes through the foci $F_1$ and $F_2$ of the ellipse, two curves intersect in $4$ points. pgfplots: correct rounding problems I had a problem with disappearing marker as the data sum up to more than 100 while xmax was set to 100. See .  As written in the answer to the cited question I tried to avoid the problem by normalizing the value to 100. But this lead to (much smaller) rounding errors in previously correct rows and now the marker disappear there. See the following example. Before the normalisation the column F ist affected, after the column B.   How can I in a reliable way avoid this rounding problem? Or at least get a clear error message. I have quite a number of this plots and I don't want to have to check them all the time for missing markers.    \documentclass{scrreprt} \usepackage{pgfplots} \pgfplotsset{compat=1.13} \usepackage{pgfplotstable} \pgfplotsset{  my stackbar plot/.style={              xbar stacked,              xmin=0,xmax=100,              symbolic y coords={A,B,C,D,E,F,G},              ytick=data,              nodes near coords={xxx},}}  \begin{document}  \pgfplotstableread[col sep=space]{  text   --  -    +       ++  A      0.0 1.7 13.8    84.5  B      0.0 0.6 20.1    79.3  C      0.0 1.9 13.2    84.9  D      0.0 1.6 27.9    70.5  E      1.3 3.9 19.5    75.3  F      0.0 1.4 15.0    83.7  G      0.3 1.7 24.8    73.2  }\data   \pgfplotstablecreatecol[create col/expr={\thisrow{--}+\thisrow{-}+\thisrow{+}+\thisrow{++}}]{sum}\data %  \pgfplotstablecreatecol[create col/copy=--]{--o}\data  \pgfplotstablecreatecol[create col/copy=-]{-o}\data  \pgfplotstablecreatecol[create col/copy=+]{+o}\data  \pgfplotstablecreatecol[create col/copy=++]{++o}\data  \pgfplotstablecreatecol[create col/expr={100/\thisrow{sum}*\thisrow{--o}}]{--}\data  \pgfplotstablecreatecol[create col/expr={100/\thisrow{sum}*\thisrow{-o}}]{-}\data  \pgfplotstablecreatecol[create col/expr={100/\thisrow{sum}*\thisrow{+o}}]{+}\data  \pgfplotstablecreatecol[create col/expr={100/\thisrow{sum}*\thisrow{++o}}]{++}\data  \pgfplotstablecreatecol[create col/expr={\thisrow{--}+\thisrow{-}+\thisrow{+}+\thisrow{++}}]{sumnew}\data  \pgfplotstabletypeset[columns={text,sum}   ,precision=10,columns/text/.style={string type}]\data \quad \pgfplotstabletypeset[columns={text,sumnew},precision=10,columns/text/.style={string type}]\data   %\pgfplotstablesave{\data}{pgfplotstempout.dat} \begin{tikzpicture} \begin{axis}[my stackbar plot]  \addplot table [x expr = \thisrow{--o},y=text] {\data};  \addplot table [x expr = \thisrow{-o}, y=text] {\data};  \addplot table [x expr = \thisrow{+o}, y=text] {\data};  \addplot table [x expr = \thisrow{++o}, y=text] {\data}; \end{axis} \end{tikzpicture} \quad \begin{tikzpicture} \begin{axis}[my stackbar plot]  \addplot table [x expr = \thisrow{--},y=text] {\data};  \addplot table [x expr = \thisrow{-}, y=text] {\data};  \addplot table [x expr = \thisrow{+}, y=text] {\data};  \addplot table [x expr = \thisrow{++}, y=text] {\data}; \end{axis} \end{tikzpicture}  \end{document} Any evidence that subdomain can help with domain authority and vice-versa? What does the 'static' keyword do in a class? To be specific, I was trying this code:  package hello;  public class Hello {      Clock clock = new Clock();      public static void main(String args[]) {         clock.sayTime();     } }   But it gave the error      Cannot access non-static field in static method main   So I changed the declaration of clock to this:  static Clock clock = new Clock();   And it worked. What does it mean to put that keyword before the declaration? What exactly will it do and/or restrict in terms of what can be done to that object? Javascript for multiple elements In a question on "along" vs "alongside", I have seen this question and the correct answer on the web site makes me confused.      The question is: I enjoyed my walk _________________ the river.  a) along b) alongside  I selected "...along the river" as the correct answer, but on the web site it is shown as wrong and the correct answer is "alongside".  How can that be? I still do not agree, because I have studied the difference between the two. "along" is just the right preposition for thin and long places like road, path, street, beach etc".   So, why would the correct answer be "alongside" which simply means "next to"? Do you not think "along" seems to be better fitting?  Regards,  Note: I would be more than gratefull, if somebody show it to me in a video, like "a walk along the river" and "a walk alongside the river". Maybe I can understand the how "a walk along the river" and "a walk alongside the river" would take place. Why doesn't my systemd user unit start at boot? Usage in sentence:  Person A: Ooh - look at that crow!  Person B: Well, actually, as someone who is a scientist who studies crows, I am telling you, specifically, in science, no one calls jackdaws crows. They're not the same thing. If you're saying "crow family" you're referring to the taxonomic grouping of Corvidae, which includes things from nutcrackers to blue jays to ravens. It's not one or the other, that's not how taxonomy works.  Person A: Are you always such an XXX?   "Pedant" is close, but not quite.  It would be a mix of "know-it-all", "condescending" and "arrogant".