Prove that solution of the equation $8x^3-4x^2-4x+1= 0$ has roots $\cos\frac{\pi}{7},\cos\frac{3 \pi}{7}\space \text{and}\space \cos\frac{5 \pi}{7}$.    How to even solve it ? I have no idea.   Because cubic have no formula like quadratic equations to obtain the roots and the expression like $\cos\frac{\pi}{7}$ is something that I don't think I can evaluate with my limited knowledge (I'm in highschool) of only $\cos 2x$ or $\cos 3x$ type. Since it has denominator of 7, I thought maybe the seventh root of unity might help me, but since only some of its multiples are used, I don't think so it will be of much use.   How do I solve this problem. Please help.
Prove that $\cos\theta$ is root of equation $8x^3-4x^2-4x+1=0$, given $\theta=\frac{\pi}{7}$.   I put $\cos\theta$ in equation, but couldn't show the left-hand side to be zero.
How to properly render smoke on a transparent background? When rendering volumes with Film > Transparent enabled, the alpha appears to be the wrong type (premultiplied/straight). In the 3D view it works as expected:    This is a known .  Is there any way to work around this? Chat requirement to post images should be made explicit There is (apparently) a 100 reputation requirement to upload and post images through chat.  There isn't any easily located documentation on this, however. 100 global reputation is listed as the privilege for , while the chat FAQ and help both do not mention any restrictions on uploading images.  As such, could somewhere other than  be used to indicate that there is a reputation requirement in chat for uploading images? Integrate $\int_0^{\pi/2} \frac{1}{1+\tan^\alpha{x}}\,\mathrm{d}x$ Evaluate the integral $$\int_0^{\pi/2} \frac{1}{1+\tan^\alpha{x}}\,\mathrm{d}x$$ Suppose $z\neq -1$ is a complex number of norm 1m $(|z| =1)$. I know that $z= n + mi$, what is the most straightforward way of solving this problem? I was also given the following sketch for a possible approach. My 5 year- old daughter is constantly defiant! My 5-year old daughter is very defiant.  It started getting bad about 6 months ago. She doesn't do what she is told, and any time she hears the word "no" she throws terrible, dramatic fits, screaming ridiculously loud, and in some cases hitting.  She refuses to make eye contact when I warn her to stop doing something even when I am down at her eye level and it takes multiple orders to finally get her to look at me. She constantly tells me no.  I feel like I constantly tell her no and that she feels this gives her the right to tell her father and I no.  I have tried time outs which are not effective at all; she only throws a fit the whole time, never seeming to run out of wind. Spankings are only effective until the shock of the action itself wears off and then she returns to the same behavior.  My husband and I recently took away ALL of her toys/ possessions being she is a pretty materialistic child (she loves getting toys and always wants something new). We told her that if she was good and listened and stopped telling us no and throwing fits, that after three days we would let her choose three boxes of toys to bring back in her room. This worked but as soon as she got those three boxes unpacked, we packed them back up because she went right back to the no's and fit-throwing within a few hours.  I am mainly the one disciplining her and I feel she is definitely the most defiant with me. I believe in explaining to her why things are not OK and why I tell her no, but I really don't feel it helps and I never feel as if she listens to what I tell her.  I am at my wits' end and feel like a failure of a mother. She is my first and only child, and the only grandchild on both sides of our family. She is an angel at school but home is completely different.  I need help figuring this out but nobody I talk to really has any advice for me. I am at the point where I am wondering if she or I need counseling because I feel out of options. Prove that the subsequence of a convergent series with nonnegative terms converges example of a non differentiable manifold Take the manifold: The graph of $|x|$ on $(-1,1)$, with the induced topology from $\mathbb{R}^2$. This is a topological manifold, which is homeomorphic to $(-1,1)$ by projection. Is it a differentiable manifold? I believe it is, because we can take an atlas with only the projection, and then we will have only one transition map which is the identity, and therefore differentiable. But I'm not sure I get the definition of a differentiable manifold right...  Thanks! Query to compare the structure of two tables in MySQL To automate the backup process of one of my MySQL databases, I would like to compare the structure of two tables (current version vs old version).  Can you think of a query that can compare two tables?  Here are some example tables that you can compare.   CREATE TABLE product_today (   pname VARCHAR(150),   price int,   PRIMARY KEY (pname) );  CREATE TABLE product_yesterday (   pname VARCHAR(150),   price int,   PRIMARY KEY (pname) );  CREATE TABLE product_2days_back (   pname VARCHAR(15),   price int,   PRIMARY KEY (pname) );   The first two tables have identical structures. The last one is different. I just need to know whether two tables have different structures or not. I'm not interested in the how they differ. What's the starting wealth for higher levels? What sources of randomness can be made with commands in Minecraft? I want to make a mini game in Minecraft were I need a random command generator for 9 commands. I need a random command to give the player a random item. I can't do the command block in the dispenser any more because it will just drop out instead of being placed. What different ways are there for generating randomness in commands, and which one would be best for my use? This question is similar to the one about , but I am specifically attempting to get randomness when using commands - the redstone solutions do not apply here. JavaFX: Undecorated Window Prove that $f'(0)$ exists and $f'(0) = b/(a - 1)$ Problem:   If $f(x)$ is continous at $x=0$, and $\lim\limits_{x\to 0} \dfrac{f(ax)-f(x)}{x}=b$, $a, b$ are constants and $|a|&gt;1$, prove that $f'(0)$ exists and $f'(0)=\dfrac{b}{a-1}$.   This approach is definitely wrong:    \begin{align} b&amp;=\lim_{x\to 0} \frac{f(ax)-f(x)}{x}\\ &amp;=\lim_{x\to 0} \frac{f(ax)-f(0)-(f(x)-f(0))}{x}\\ &amp;=af'(0)-f'(0)\\ &amp;=(a-1)f'(0) \end{align}  I will show you a case why this approach is wrong:     \[f(x)= \begin{cases} 1,&amp;x\neq0\\ 0,&amp;x=0 \end{cases}\]   $\lim_{x\to0}\dfrac{f(3x)-f(x)}{x}=\lim_{x\to0} \dfrac{1-1}{x}=0$   but $\lim_{x\to0}\dfrac{f(3x)}{x}=\infty$,$\lim_{x\to0}\dfrac{f(x)}{x}=\infty$   Does anyone know how to prove it? Thanks in advance! Overriding !important style Title pretty much sums it up.  The external style sheet has the following code:  td.EvenRow a {   display: none !important; }   I have tried using:  element.style.display = "inline";   and  element.style.display = "inline !important";   but neither works. Is it possible to override an !important style using javascript.  This is for a greasemonkey extension, if that makes a difference. Why doesn't this division work in Python? 'POCO' definition Can someone define what exactly 'POCO' means? I am encountering the term more and more often, and I'm wondering if it is only about plain classes or it means something more? Is it legal to add your publications to ResearchGate? I recently created a profile on . And to add your publications list it requires you to upload the paper. Is it legal to upload published journal articles on public servers, where everyone can download it for free? I want to make a generic class that accepts only serializable classes, can it be done with the where constraint?  The concept I'm looking for is this:  public class MyClass&lt;T&gt; where T : //[is serializable/has the serializable attribute] Where can I find Vagrant packages? is a tool to build and manage virtual machines for developers.  It has a  guide that contains iffy-looking suggestions like  $ sudo ln -s /usr/bin/ruby1.8 /usr/bin/ruby  # wtf???   or installing RubyGems from source rather than from Ubuntu packages, and then using gem install to install vagrant itself.  I'm not feeling comfortable just following those instructions.  Is there perhaps a PPA?  Are there, perhaps, alternative tools that are packaged for Ubuntu? I was able to post in chat with less than 20 rep by joining 20 sites I joined 20 other sites (out of the many) and even with 1 rep was able to post in any of the chatrooms. For example: In ELL, I have just 1 rep and have posted messages in its chat.  ell.stackexchange.com/users/18923/lamart This seems like a bug because I should have needed 20 rep to post. . EDIT: From , I got the idea to make this edit. To reproduce this bug/error:  Create a new SE account. Use it to make an account here. You will get 1 default rep. Choose a random SE site and use this account to get 1 rep on that random site. Repeat the above step 20 times. Now you have 1 rep on 20 network sites but the net SE rep is 20. Go to the ELL chat and ping me. (important step)  From my comment: I have now got 40 rep since people voted this question up. Initially I had net 20 rep gained by creating 20 accounts on 20 different SE sites. Those default 1rep added up to get me to 20rep and I was able to chat. The link says one needs 20 rep in any of the SE site to chat in &quot;all the other&quot; category chatrooms..... but I did not have 20 rep on ELL at that moment. ${{2\pi i=0}}$? I was trying to solve for $i^i$.  I got distracted and did the following: $e^{i \theta} = \cos(\theta) + i\sin(\theta)$ $e^{i 2\pi} = cos(2\pi) + i\sin(2\pi) = 1$ ||| Take ln() of two sides of equation: $\ln(e^{i2\pi}) = \ln(1)$ $i2\pi = 0$ Clearly this is false. What did I do wrong?