It is to be solved using trigonometry. I tried taking $x$, $y$ and $z$ as $\sin(a)$, $\sin(b)$ and $\sin(c)$ respectively, but could get no further.
If $xy + yz + zx = 1$, then show that      $$\dfrac{x}{1-x^2} + \dfrac{y}{1-y^2} + \dfrac{z}{1-z^2} = \dfrac{4xyz}{(1-x^2)(1-y^2)(1-z^2)}$$   I tried doing the sum algebraically, that is, by solving the LHS but, that method isn't really getting us anywhere.  Also, I found this sum in an a book on trigonometry. So how can I solve it with trigonometry?
My try, using $x = \sec(u)$ substitution:  $$ \begin{eqnarray} \int \frac{\sqrt{x^2-1}}{x} \mathrm{d}x &amp;=&amp; \int \frac{\sqrt{\sec^2(u) - 1}}{\sec(u)}\tan(u)\sec(u) \mathrm{d}u \\ &amp;=&amp; \int \tan^2(u) \mathrm{d}u \\ &amp;=&amp; \tan(u) - u + C \\ &amp;=&amp; \tan(arcsec(x)) - arcsec(x) + C \end{eqnarray} $$  However, according to , the answer should be: $$ \int \frac{\sqrt{x^2-1}}{x} \mathrm{d}x = \sqrt{x^2-1}+\arctan \left( \frac{1}{\sqrt{x^2-1}} \right)+C $$ When I derive this last answer I don't get back the integrand, but rather: $$ \frac{\mathrm d}{\mathrm d x}\left(\sqrt{x^2-1}+\arctan \left( \frac{1}{\sqrt{x^2-1}} \right)+C\right) = \frac{x}{\sqrt{x^2-1}}- \frac{x}{(x^2-1)^{3/2}\left(1+\frac{1}{x^2-1}\right)} $$  I don't know how to simplify this expression more. Also, I am unable to check whether my answer is correct because I don't know how to find the derivative of $arcsec(x)$.  Can someone check my calculations and tell me where I've done something wrong and how one can simplify the last expression to get back the integrand? How would I go about finding $\prod_{n=1}^\infty (1+\frac{1}{n^2})$ ? Setting $a_k = \prod_{n=1}^k (1+\frac{1}{n^2})$, the first few partial products are $a_1 = 2$, $a_1 = 2.5$, $a_3 \approx 2.78$, $a_4 \approx 2.591$, $a_{20} \approx 3.50$, and they eventually appear to converge at around $3.639$. (These values were found with a quick python program. So much for my initial conjecture that the product converged to $e$.)   I tried writing $1+\frac{1}{n^2}$ as $\frac{n^2 + 1}{n^2}$, but it wasn't immediately obvious how to prove convergence / find the limit because the product doesn't really factor or telescope. Is there a way to calculate the limit? Is time continuous or discrete? I'm having some trouble updating a row in a MySQL database. Here is the code I'm trying to run:  import MySQLdb  conn=MySQLdb.connect(host="localhost", user="root", passwd="pass", db="dbname") cursor=conn.cursor()  cursor.execute("UPDATE compinfo SET Co_num=4 WHERE ID=100") cursor.execute("SELECT Co_num FROM compinfo WHERE ID=100") results = cursor.fetchall()  for row in results:     print row[0]  print "Number of rows updated: %d" % cursor.rowcount  cursor.close() conn.close()   The output I get when I run this program is:     4Number of rows updated: 1   It seems like it's working but if I query the database from the MySQL command line interface (CLI) I find that it was not updated at all. However, if from the CLI I enter UPDATE compinfo SET Co_num=4 WHERE ID=100; the database is updated as expected.  What is my problem? I'm running Python 2.5.2 with MySQL 5.1.30 on a Windows box. How to make Gnome's "Activities" bar always visible? Reinstall Windows 8 on a Windows 8 based system with embedded key I'm a computer technician and we started selling few Windows 8 machine. We saw the new COA label does not show the key and after reading a little bit online, realized those systems came with the key emmebeded in the eUFI BIOS.   My question is : How do I reinstall Windows 8 if the hard drive is crashed? I kind of get that you can install it with a media and the media should activate the licence embedded. What is the media required? Is a retail DVD is able to do the work or do I need a system builder kit? I understand the manufacturer is reponsible to provide media  or replace the failed drive under the warranty but what will happen when warranty expire? I'll have to sell a full windows 8 licence with a key?  Thanks prevent user creating a duplicate node title how to prevent user creating a duplicate node title for a content type?  I already try uniqueness module () but it's only list out the duplicate content title but not return any error message when user save the new content. Any suggestion module or code? :) Exporting coordinates from LAS files to CSV? Is there any tool available for extracting the X, Y and Z coordinates from a .las file and exporting them to a .csv or similar file? out file with multiple criteria in python? Dynamically update a graphic object in Mathematica We are using .Net to make web service calls the Salesforce API. Is there a way to know for sure that the calls are using TLS 1.2? Are there any API logs or records of the protocol is use in Salesforce we can review to be confident the change is working? I've made some high resolution pixel art in PS by drawing it down at the pixel level and then increasing the image size with the nearest neighbor setting to keep it crisp. In the web browser, however, I need to display these images as icons with a dynamic size, so I can't just size the images correctly to begin with, and the broswer seems to blur the images when rendering them smaller, while vector images seem to stay more crisp as they shrink.   So I'm wondering if there's a way to convert the pixel art neatly into vector. I'm aware of the trace function in illustrator, but it seems to do a rough job iirc. Diagonalizable Operators and characteristic polynomials Game resource timers - general implementation I'm interested in understanding how resource generation timers work on many of the mmorts type games out there. I can't wrap my head exactly around how the timers are synched with servers, especially with multiple users. If I need to provide more details, please let me know.  Thanks! Using ArcPy to convert .txt to shapefile with coordinate system? I tried to define my coordinate system on creating the shapefile but when I zoom to layer I don't see my features and ArcGIS crashes. I know it's not in the coordinate system I defined for it but it shows the coordinate under properties but doesn't reflect when I zoom to layer. Here is my python code:  import arcpy  newfcName ="newfc.shp"   outpath = r"C:/FLOOD/IKEJA AND KOSOFE"  # Declaration arcpy.env.overwriteOutput=True  arcpy.env.workspace= outpath  # Create new Shapefile and add FIELDS newfc = arcpy.CreateFeatureclass_management(outpath, newfcName, "Point") # get the coordinate system by describing a feature class dsc = arcpy.Describe("newfc.shp" )  coord_sys = dsc.spatialReference  try:        # run the tool    arcpy.DefineProjection_management(newfcName , coord_sys)     # print messages when the tool runs successfully    print(arcpy.GetMessages(0))  except arcpy.ExecuteError:    print(arcpy.GetMessages(2))  except Exception as ex:    print(ex.args[0])  arcpy.AddField_management(newfc, "X", "STRING", field_length = 50) arcpy.AddField_management(newfc, "Y", "STRING", field_length = 50) arcpy.AddField_management(newfc, "Z", "STRING", field_length = 50) arcpy.AddField_management(newfc, "I", "FLOAT", field_length = 50)  # Reference Cursors cursor=arcpy.da.InsertCursor(newfc, ["X", "Y", "Z", "I"])  # Read File  a = open("C:/FLOOD/IKEJA AND KOSOFE/DEM/532722.txt","r")  inputF = a.readlines()  for line in inputF:     xCoordinate, yCoordinate, zValue, iValue = line.split(" ")      newRow = (str(xCoordinate), str(yCoordinate), str(zValue), float(iValue))      cursor.insertRow(newRow)  a.close() Cardinality of the set of all real functions of real variable How does one compute the cardinality of the set of functions $f:\mathbb{R} \to \mathbb{R}$ (not necessarily continuous)? Some conditions require that a person lie in bed all the time until recovered. Is there a name for such conditions or people experiencing them at the moment?   For example,      I am currently &lt;bed sick&gt; and can't help you with your homework.   I tried looking for bed sick but hit a dead end. Suppose 27 cubes are stacked together, suspended in the air, to form a larger cube.  The cube is then painted on all the exposed surfaces and dried.  The smaller cubes are then randomly permuted in both spatial position and spatial orientation to form another large cube.  What is the probability that new larger cube is identical to the original?  This is not homework, it was passed on by a friend who found it somewhere on the internet, and they do not remember where. Mail App keeps popping up in the background in Mac OS Mojave Is there a way to search the latest version of every file in TFS for a specific string or regex? This is probably the only thing I miss from Visual Source Safe...    Currently I perform a Get Latest on the entire codebase and use Windows Search, but this gets quite painful with over 1GB of code in 75,000 files.    EDIT: Tried the powertools mentioned, but the "Wildcard Search" option appears to only search filenames and not contents.  UPDATE: We have implemented a customised search option in an existing MOSS (Search Server) installation.