Using trigonometric identities to simply the following expression $\tan\frac{\pi}{5} + 2\tan\frac{2\pi}{5}+ 4\cot\frac{4\pi}{5}=\cot\frac{\pi}{5}$
To prove $ \tan(A) + 2 \tan(2A) +4\tan4A + 8 \cot8A =\cot(A) $
Subscribe to a particular question by email (not RSS) How can I permanently enable cheating? Let $a&gt;0$ and $f: \mathbb R \rightarrow \mathbb R$ with a property $\forall x,y \in \mathbb R;$ such that $0\;&lt;\;y\;-\;x\;&lt;\;a$ $$f(x)\;&lt;f(y).$$ Prove if the statements are true or false:  $1)$ $f$ must be injective.  $2)$ $f$ must be surjective.  I tried to use a quarter of a circle with radius $r=a$ or a right-angled triangle and translate it. When I moved it, I saw new values of $f$. I concluded the maximum value in the previous interval can't appear again in another interval when the distance between one point in the first interval and one point in the second is less than $a$. I also considered using an absolute value function, but I haven't done anything. I think, but I haven't proven it, that the function must be bijective. Often (via Google Now, or when searching for a solution to some problem), I'll open a question from a non-favourite site in the app. Sometimes, I'd like to see what else is going on in the site where the question was posted. Currently, I have one of two ways:   Via tag icon:   Click one of the tag icons beneath the title Delete the [tag] text (at which point, thankfully, the sorting changes to Active from Votes).  Via manually searching for the site, which is more cumbersome. I type in hard and Hardware Recommendations is the fourth entry (below Ask Different, Raspberry Pi and Arduino)! This more annoying for meta sites, where I have to pick Meta from the dropdown menu.   Can I pretty please have a way to head to the site directly from the question? Maybe:   the site name in the top bar could become clickable and lead to the site, or an option in the More menu?     This would be especially useful for Meta sites, where questions posted around the same time could be related. Interestingly enough, most questions that Google Now are for metas of sites I visit on occasion, but haven't pinned to the list. What the #$@&%*! is that called? Lucia is gone. Just gone Skyrim problem here. I've finally found a house mod that will support a family (as in allow them to move there). I'm excited to install it, but it says that a new save game is highly recommended unless you haven't adopted any kids yet.   I recently adopted Lucia. I have two saves in my game, a quick and a full. I know that the full save has Lucia as adopted, but I'm not sure if the quick does. If it does not, then my reasoning is that I can install the mod and load that save, rather than having to work all the way back through eighteen levels.   And here's the problem. I loaded the quick save and searched for Lucia, in an attempt to see if she was adopted or not. She is nowhere to be found. She is not sitting under the tree in the middle of Whiterun, she is not in Breezehome, and she is not in the inn.   I tried to use console commands. both the moveto and placeatme commands do not work. I used both the refid and the baseid. Neither worked. I used help Lucia, thinking maybe the wiki was wrong on her ID. it gave me a different ID, which also didn't work. Every time, the console says either that it can't find the objectbaseid, or that the id is invalid.   This has led me to the conclusion that Lucia has been entirely erased from the game's memory, and I KNOW that hasn't happened. Can someone tell me what is going on?   Additional details: I am using a Breezehome overhaul mod. When I adopted Lucia, she moved in perfectly with no trouble though, so I can't really see how that would be the problem. Who is the Community user? Size in bytes is significantly less ( half the size) then size in MB in get info window In the get info window it says 387,037,139 bytes which is 387 MB whereas the size on disk is 716.9MB which is 716,900,000 bytes.   du -sh paramgen  result (in MiB) 684M    paramgen  Why is there such a large difference? GParted can't resize partition, but unallocated partition is available Is there a word or phrase for something that one might wish exists, but most certainly doesn't? Do you need to take the Hide action when you are Invisible, make no noise and leave no traces? The Situation: Lets say you are under the effect of Invisibility.  An invisible creature is impossible to see without the aid of magic or a special sense. For the purpose of hiding, the creature is heavily obscured. The creature's location can be detected by any noise it makes or any tracks it leaves.  And you are wearing the Boots of Elvenkind  While you wear these boots, your steps make no sound, regardless of the surface you are moving across. You also have advantage on Dexterity (Stealth) checks that rely on moving silently.  Now your groups ranger or druid casts Pass Without Trace on you.  [...] can’t be tracked except by magical means. A creature that receives this bonus leaves behind no tracks or other traces of its passage.  My Question(s): I know that when you are Invisible and you did not hide, you are still detectable because of the noise and the track you leave behind. But the Boots of Elvenkind and Pass Without Trace take care of that. So:  Are you still required to take the hide action in this situation if you don't want enemies to know your location/presence? If so, why? What gives away your location? If not and you are under the effect of greater Invisibility do you get the benefits of &quot;Unseen Attackers and Targets&quot; for every attack you make when you move between attacks? (You would give away your location, but are not detectable, so completely hidden again when you move) String representation of an Enum How old is SpongeBob? I don't mean how old is the series, I mean what is SpongeBob's age if he were a real person. I ask because his age seems to be ambiguous in the show.   He has a full-time job He has his own house He is always referred to as a kid (especially in the movie) He acts childish and immature In one episode, he was pretending to be an adult/grown-up How to replace text in the body of an incoming message with a hyperlink in Outlook 2010? How can I remove myself from India's visa blacklist? How can I find $\lim_{n\to \infty} a_n$ Let $$a_n=\left(1-\dfrac{1}{\sqrt2}\right)\dots \left(1-\dfrac{1}{\sqrt{n+1}}\right),n\ge1$$ Then find $\lim_{n\to \infty} a_n$.   How can I proceed? I am stuck at the first step. Please help. How to compute $\lim _{x\to 0}\frac{x\bigl(\sqrt{3e^x+e^{3x^2}}-2\bigr)}{4-(\cos x+1)^2}$? I have a problem with this limit, I don't know what method to use. I have no idea how to compute it. Is it possible to compute this limit with the McLaurin expansion? Can you explain the method and the steps used? Thanks. (I prefer to avoid to use L'Hospital's rule.)  $$\lim _{x\to 0}\frac{x\bigl(\sqrt{3e^x+e^{3x^2}}-2\bigr)}{4-(\cos x+1)^2}$$ Sort file by group of lines If I have a file with contents similar to:  FirstSection     Unique first line in first section     Unique second line in first section  SecondSection     Unique first line in second section     Unique second line in second section  ...  NthSection     Unique first line in Nth section     Unique second line in Nth section   Is it possible to use unix commands (e.g. sort, awk) to sort the file alphabetically by the first non-indented line in each three line group, whilst keeping the indented lines under their existing group? Can the Ubuntu logo be used for a commercial website? Is there an Ubuntu logo that can be used on a commercial website to indicate that Ubuntu can be installed for customers? In Cocoa KVO, why doesn't a change on a NSMutableArray proxy notify observers?