Please give me full proof. I am expected to prove by induction. I tried all sort of things but I got stuck at $$ \frac{s_1..s_{n-1}+\tan A_n+\tan A_n(-s_2+s_4....s_{n-1})}{1-s_2+s_4...s_{n-2} - \tan A_n(s_1-s_3...s_{n-1}} $$  Where $s_r$ is sum of products of tangents of angles $A_1,A_2,...A_n$ taken $r$ at a time
$$\tan(\theta_1+\theta_2+\cdots+\theta_n)=\frac{S_1-S_3+S_5+\cdots}{1-S_2+S_4+\cdots}$$  where  $S_i$ denotes the sum of product of tangent of angles taken $i$ at a time.     For example,$$\tan(\theta_1+\theta_2)=\frac{S_1}{1-S_2}=\frac{\tan\theta_1+\tan\theta_2} {1-\tan\theta_1\tan\theta_2}$$   (This formula is given  in my textbook with no derivation or background)  How to derive this?
Ways to give users some specific education about question quality and topicality Important: This question is being asked here, and not on the Programmers Meta,  because I believe that several SE sites suffer from this same, specific problem.    On Programmers, I see a pattern emerging.   New User sees the "Programmers" site title, and asks their incomplete, "fix my broken code" question. Community members comment that their question is off-topic, and recommend Stack Overflow.  New User says "thanks" and leaves, leaving their undeleted question for the community to clean up.    What seems clear to me is that:   New User has never seen the "What kinds of questions can I ask about here" page, or if they have, they've clicked through it without reading it. New User doesn't know how to move their question to the right place properly  (i.e. post on new site, delete from old site). Community cleanup of such questions is onerous.  The moderators on Programmers, following the principle that the community should moderate itself for the most part, are not proactive about removing such questions unless they're especially egregious. Migration is even more onerous.   OK, so we have the .  Here's what it says about topicality, about halfway down the page:    Note to those who are confused about Programmers' site scope: it's all right there, in black and white.  However, I do notice a problem.  It's circled in red.  That's our fault, the fault of the Programmers community.  What we really meant to say was "Don't ask your code troubleshooting questions here; those belong on Stack Overflow."  Coding tools deserves its own bullet.  We should fix that.  However, what are the chances that the user is actually seeing this, and evaluating whether or not they should ask their question based on this?  So here's my question, in two acts:  ACT I:  Is there a way to highlight the pain points of a particular site to new users specifically, so that we can be very clear that we don't want those questions that are clearly and unambiguously off-topic on our site, before they ask their question?  ACT II: Failing that, is there a way that we can fast-track the removal of such questions so that they no longer pollute our front page? $\lfloor x\rfloor + \lfloor y\rfloor \leq \lfloor x+y\rfloor$ for every pair of numbers of $x$ and $y$ Give a convincing argument that $\lfloor x\rfloor + \lfloor y\rfloor \leq \lfloor x+y\rfloor$ for every pair of numbers $x$ and $y$. Could someone please explain how to prove this? I attempted to say that the largest values that could be added to $x$ and $y$ is $0.99$ and that doing so still made $\lfloor x\rfloor + \lfloor y\rfloor &lt; \lfloor x+y\rfloor$. However, my answer was not accepted. How do I challenge an audit failure? I've run through Both SO and Meta searches for an answer, as well as reviewing the , and I cannot find a canonical process to challenge an audit failure.  I can find discussions of the single-choice "I Understand" topic that appear unresolved, in that no solution presently exists.  The particular question in point is .  Despite the number of up-votes, I feel that this quite clearly falls into a common closure category: "Here's my problem; how do I write the program?"  Compare this to   I also failed review on the first link I posted above, .  This looks like a case of code review, rather than SO.  I don't want to get rid of audits; I've learned good things from some of my failures, especially at the start.  However, in the past few weeks, a majority of my failures have been highly questionable, and I've drawn more than one suspension from those that are outright wrong by current standards.  If we have a challenge mechanism, please point me in the right direction.  Otherwise, it's simply not worth the annoyance any more, and I'll drop back to merely comments and answers. Can I put 700c tires on rims made for 27 x 1.25"? How to reset Canon 70D WiFi? Canadian visa rules for flight connections I know that in the US foreigners must have a US visa if they have a connection in the US, even if they fly from a different country to a different country. That is because in the US you have to go through passport control regardless of your final destination.  What is the situation in Canada? Does a person who would need a visa to enter Canada need a visa to connect through Canada without exiting the airport? No Badge Progress color for Golden Badges? Transpose/Unzip Function (inverse of zip)? I have a list of 2-item tuples and I'd like to convert them to 2 lists where the first contains the first item in each tuple and the second list holds the second item.  For example:  original = [('a', 1), ('b', 2), ('c', 3), ('d', 4)] # and I want to become... result = (['a', 'b', 'c', 'd'], [1, 2, 3, 4])   Is there a builtin function that does that? How to handle UNABLE_LOCK_ROW errors when this is caused by a batch APEX job we are facing a serious issue. We have a batch APEX job running daily which makes updates on records fulfilling certain conditions. There is another job running parallel to this, in some cases updating or accessing to the same records than the batch job and hence, causing the error:     Update failed. First exception on row 400 with id XXX; first error:    UNABLE_TO_LOCK_ROW, unable to obtain exclusive access to this record or 200    records:   So, I had then read that in order to fix this kind of error you should use FOR_UPDATE in your select statement (records that have to be updated). The problem is, that you can not use FOR UPDATE for batch jobs. We tested this and effectively, we get the error:     System.QueryException: Locking is implied for each batch execution and   therefore FOR UPDATE should not be specified      It will however lock the records within each execution of execute. See   QueryLocator.getQuery()usage:      You cannot use the FOR UPDATE keywords with a getQueryLocator query to   lock  a set of records. The start method automatically locks the set   of records in  the batch.   Could you find a solution to this?. We would like to handle this in the code itself, because we don't have the permission to schedule the batch job for another time as the another job. Is const_cast safe? I can't find much information on const_cast.  The only info I could find (on Stack Overflow) is:     The const_cast&lt;&gt;() is used to add/remove const(ness) (or volatile-ness) of a variable.   This makes me nervous.  Could using a const_cast cause unexpected behavior?  If so, what?  Alternatively, when is it okay to use const_cast? On the moderator , the tooltip will display the text used for voting on a question; this is incorrect.   Steps to reproduce:   Cast a vote for or against any candidate Undo that vote   At this point, the tooltip content is changed, at least for that candidate.  See screenshots:           As you can see, the message doesn't quite fit here; these are certainly not questions. sum of polynoms of given property I have to figure out a way to count how many sequences are there s.t:   My alphabet is $1$ up to $49$. Each number that is chosen, is chosen only once. The sequence is $6$ digits long. The sequence is in increasing order.  e.g that is a valid sequence: $(1, 2, 40, 42, 43, 49)$  and this is not: $(1, 2, 40, 25, 25, 12)$   I know that for the first 3 rules the answer is: $\binom{49}{6}\cdot 6!$  How do I deal with the fourth? How do you address unknown peers in email? This is one bit of academic etiquette I'm not sure about. When I write to another researcher, a peer (we're both professors). I've never exchanged with before and never met him but I know of him (and he probably knows of me).  Erring on the side of formality, I write to him “Dear professor Mortimer”. I sign with my first name, because that's how I've always signed my emails (there's a formal signature block below, with full name and contact details) and I think it indicates a willingness to be called by my first name (which I prefer, but I don't want to force people one way or the other).  Is that going to be well received? Will people take the hint? Or should I just start using first names from the start? (it does sound weird to write “dear Philipp” to someone you don't know) Is it true that: $a+bi$ is prime in $\mathbb{Z}[i]$ if and only if $a^2+b^2$ is prime in $\mathbb{Z}$ Why are these pistons pushing? I've lately become hooked into building computers in Minecraft and today I decided to build a giant hard drive. I've built HDDs before, but not at the scale I'm currently attempting, and I anticipated problems. As I was wiring some pistons, they activated and pushed forward oddly. As far as I can tell, there's nothing powering them.    I thought at first that it might be the redstone torches (top left of above pic) but they aren't powered.   Here's a top view of the same pistons. I also thought the line of redstone above  them might be the problem (middle of above pic), but when I destroy it, the pistons stay extended.  Is there a reason why my pistons are extending? If so, is there a way to get them to power (when I want them to) without disturbing the pistons above? Boundary of Ball of radius R has zero measure PDF Editing in PHP? Does anyone know of a good method for editing PDFs in PHP? Preferably open-source/zero-license cost methods. :)  I am thinking along the lines of opening a PDF file, replacing text in the PDF and then writing out the modified version of the PDF?  I have programmatically created PDF files in the past using FPDF, but found it a little unwieldy at times. Prove every element of $G$ has finite order. Find $f$ if $$ f(x)+f\left(\frac{1}{1-x}\right)=x $$ I think, that I have to find x that $f(x) = f\left(\frac{1}{1-x}\right)$ I've tried to put x which make $x = \frac{1}{1 - x}$, but this equation has no roots in real numbers.